THE THEORY OF MACHINES
THE NATURE OF THE MACHINE
To understand the nature of the machine, detail of the most common and best known machines, the reciprocating engine are given first. It consists of the following essential, independent parts:
(a) The part which is rigidly fixed to a foundation or the frame- work of a ship, and which carries the cylinder, the crosshead guides, if these are used, and at least one bearing for the crank- shaft, these all forming parts of the one rigid piece, which is for brevity called the frame, and which is always fixed in position.
(6) The piston, piston rod and crosshead, which are also parts of one rigid piece, either made up of several parts screwed together as in large steam and gas engines, or of a single casting as in automobile engines, where the piston rod is entirely omitted and the crosshead is combined with the piston. It will be convenient to refer to this part as the piston, and it is to be noticed that the piston always moves relatively to the frame with a motion of translation, 1 and further always contains the wristpin, a round pin to facilitate connection with other parts. The pis- ton then moves relatively to the frame and is so constructed as to pair with other parts of the machine such as the frame and connecting rod now to be described,
(c) The connecting rod is the third part, and its motion is peculiar in that one end of it describes a circle while the other end, which is paired with the wristpin, moves in a straight line, which latter motion is governed by the piston. All points on the rod move in parallel planes, however, and it is said to have plane motion, as has also the piston. The purpose of the rod is to transmit the motion of the piston, in a modified form, to the remaining part of the machine, and for this purpose one end of it is bored out to fit the wristpin while the other end is bored out to fit a pin on the crank, which two pins are thus kept a fixed distance apart and their axes are always kept parallel to one another,
(d) The fourth and last essential part is the crank and crankshaft, or, as it may be briefly called, the crank. This part also pairs with two of the other parts already named, the frame and the connect- ing rod, the crankshaft fitting into the bearing arranged for it on the frame and the crankpin, which travels in a circle about the crankshaft, fitting into the bored hole in the connecting rod available for it. The stroke of the piston depends upon the radius of the crank or the diameter of the crankpin circle, and is equal to the latter diameter in all cases where the direction of motion of the piston passes through the center of the crankshaft. The flywheel forms part of the crank and crankshaft.
In many engines there are additional parts to those mentioned, steam engines having a valve and valve gear, as also do many internal-combustion engines, and yet a number of engines have no more than the four parts mentioned, so that these appear to be the only essential ones.
Another well-known machine, is is the lathe. All lathes contain a fixed part or frame or
bed which holds the fixed or tail center, and which also contains
bored bearings for the live center and gearshafts. Then there is
the live center which rotates in the bearings in the frame and which
drives the work, being itself generally operated by means of a
belt from a countershaft. In addition to these parts there is the
carriage which holds the tool post and has a sliding motion along
the frame, the gears, the lead screw, belts and other parts, all
of which have their known functions to perform, the details of
which need not be dwelt upon.
4. Parts of the Machine. These two machines are typical of
a very large number and from them the definition of the machine
may be developed. Each of these machines contains more than
one part, and in thinking of any other machine it will be seen
that it contains at least two parts : thus a crowbar is not a machine,
neither is a shaft nor a pulley; if they were, it would be difficult
to conceive of anything which was not a machine. The so-called
" simple machines," the lever, the wheel and axle, and the wedge
cause confusion along this line because the complete machine
is not inferred from the name: thus the bar of iron cannot be
called a lever, it serves such a purpose only when along with it
is a fulcrum; the wheel and axle acts as a machine only when it
is mounted in a frame with proper bearings; and so with the
wedge. Thus a machine consists of a combination of parts.
5. Again, these parts must offer some resistance to change of
shape to be of any value in this connection. Usually the parts
of a machine are rigid, but very frequently belts and ropes are
used, and it is well known that these serve their proper purpose
only when they are in tension, because only when they are used
in this way do they produce motion since they offer resistance
to change of shape. No one ever puts a belt in a machine in a
place where it is in compression. Springs are often used as in
valve gears and governors, but they offer resistance wherever
used. Thus the parts of a machine must be resistant.
6. Relative Motion. Now under the preceding limitations a
ship or building or any other structure could readily be included,
and yet they are not called machines, in fact nothing is a machine
in which the parts are incapable of motion with regard to one
another. In the engine, if the frame is stationary, all the other
parts are capable of moving, and when the machine is serving
its true purpose they do move; 'in a bicycle, the wheels, chain,
pedals, etc., all move relatively to one another, and in all machines
the parts must have relative motion. It is to be borne in mind
that all the parts do not necessarily move, and as a matter of
fact there are very few machines in which one part, which is
referred to briefly as the frame, is not stationary, but all parts
must move relatively to one another. If one stood on the frame
of an engine the motion of the connecting rod would be quite
evident if slow enough; and if, on the other hand, one clung to
the connecting rod of a very slow-moving engine the frame would
appear to move, that is, the frame has a motion relative to the
connecting rod, and vice versa.
7. In a bicycle all parts move when it is going along a road,
but still the different parts have relative motion, some parts
moving faster than others, and in this and in many other similar
cases, the frame is the part on which the rider is and which has
no motion relative to him. In case of a car skidding down a
hill, all parts have exactly the same motion, none of the parts
having relative motion, the whole acting as a solid body.
8. Constrained Motion. Now considering the nature of the
motion, this also distinguishes the machine. When a body moves
in space its direction, sense and velocity depend entirely upon the
forces acting on it for the time being, the path of a rifle ball
depends upon the force of the wind, the attraction of gravity,
etc., and it is impossible to make two of them travel over exactly
the same path, because the forces acting continually vary; a
thrown ball may go in an approximately straight line until struck
by the batter when its course suddenly changes, so also with a ship,
that is, in general, the path of a free body varies with the
external forces acting upon it. In the case of the machine, how-
ever, the matter is entirely different, for the path of each part is
predetermined by the designer, and he arranges the whole machine
so that each part shall act in conjunction with the others to
produce in each a perfectly defined path.
Thus, in a steam engine the piston moves in a straight line
back and forth without turning at all, the crankpin describes a
true circle, each point on it remaining in a fixed plane, normal
to the axis of the crankshaft during the rotation, while also the
motion of the connecting rod, although not so simple is perfectly
definite. In judging the quality of the workmanship in an
engine one watches to see how exact each of these motions is
and how nearly it approaches to what was intended ; for example,
if a point on the crank does not describe a true circle in a fixed
plane, or the crosshead does not move in a perfectly straight line
the engine is not regarded as a good one.
The same general principle applies to a lathe; the carriage
must slide along the frame in an exact straight line and the spindle
must have a true rotary motion, etc., and the lathe in which these
conditions are most exactly fulfilled brings the highest price.
These motions are fixed by the designer and the parts are
arranged so as to constrain them absolutely, irrespective of the
external forces acting; if one presses on the side of the crosshead
its motion is unchanged, and if sufficient pressure is produced
to change the motion the machine breaks and is useless. The
carriage of the lathe can move only along the frame whether
the tool which it carries is idle or subjected to considerable
force due to the cutting of metal; should the carriage be pushed
aside so that it would not slide on the frame, the lathe would be
stopped and no work done with it till it was again properly
adjusted. These illustrations might be multiplied indefinitely,
but the reader will think out many others for himself.
This is, then, a distinct feature of the machine, that the relative
motions of all parts are completely fixed and do not depend in any
wajr upon the action of external forces. Or perhaps it is better
to say that whatever external forces are applied, the relative
paths of the parts are unaltered.
9. Purpose of the Machine. There remains one other matter
relative to the machine, and that is its purpose. Machines
are always designed for the special purpose of doing work. In
a steam engine energy is supplied to the cylinder by the steam
from the boiler, the object of the engine is to convert this energy
into some useful form of work, such as driving a dynamo or
pumping water. Power is delivered to the spindle of a lathe
through a belt, and the lathe in turn uses this energy in doing
work on a bar by cutting a thread. Energy is supplied to the
crank on a windlass, and this energy, in turn, is taken up by the
work done in lifting a block of stone. Every machine is thus
designed for the express purpose of doing work.
10. Definition of the Machine. All these points may now be
summed up in the form of a definition: A machine consists of
resistant parts, which have a definitely known motion relative
to each other, and are so arranged that a given form of available
energy may be made to do a desired form of work.
11. Imperfect Machines. Many machines approach a great
state of perfection, as for example the cases quoted of the steam
engine and the lathe, where all parts are carefully made and the
motions are all as close to those desired as one could make them.
But there are many others, which although commonly and
correctly classed as machines, do not come strictly under the
definition. Take the case of the block and tackle which will
be assumed as attached to the ceiling and lifting a weight.
In the ideal case the pulling chain would always remain in a given
position and the weight should travel straight up in a vertical
line, and in so far as this takes place the machine may be con-
sidered as serving its purpose, but if the weight swings, then
motion is lost and the machine departs from the ideal conditions.
Such imperfections are not uncommon in machines; the endlong
motion of a rotor of an electrical machine, the "flapping" of a
loose belt or chain, etc., are familiar to all persons who have seen
machinery running; and even the unskilled observer knows that
conditions of this kind are not good and are to be avoided where
possible, and the more these incorrect motions are avoided, the
more perfect is the machine and the more nearly does it comply
with the conditions for which it was designed.
DIVISIONS OF THE SUBJECT
12. Divisions of the Subject. It is convenient to divide the
study of the machine into four parts:
1. A study of the motions occurring in the machine without
regard to the forces acting externally; this study deals with the
kinematics of machinery.
2. A study of the external forces and their effects on the
parts of the machine assuming them all to be moving at uniform
velocity or to be in equilibrium; the balancing forces may then
be found by the ordinary methods of statics and the problems
are those of static equilibrium.
3. The study of mechanics of machinery takes into account the
mass and acceleration of each of the parts as well as the external
4. The determination of the proper sizes and shapes to be
given the various parts so that they may be enabled to carry
the loads and transmit the forces imposed upon them from
without, as well as from their own mass. This is machine design,
a subject of such importance and breadth as to demand an en-
tirely separate treatment, and so only the first three divisions
are dealt with in the present treatise.
KINDS OF MOTION
13. Plane Motion. It will be best to begin on the first division
of the subject, and to discuss the methods adopted for obtaining
definite forms of motion in machines. In a study of the steam
engine, which has already been discussed at some length, it is
observed that in each moving part the path of any point always
lies in one plane, for example, the path of a point on the crankpin
lies on a plane normal to the crankshaft, as does also the path
of any point on the connecting rod, and also the path of any
point on the crosshead. Since this is the case, the parts of an
engine mentioned are said to have plane motion, by which state-
ment is simply meant that the path of any point on these parts
always lies in one and the same plane. In a completed steam
engine with slide valve, all parts have plane motion but the
governor balls, in a lathe all parts usually have plane motion,
the same is true of an electric motor and, in fact, the vast majority
of the motions with which one has to deal in machines are plane
14. Spheric Motion. There are, however, cases where different
motions occur, for example, there are parts of machines where a
point always remains at a fixed distance from another fixed point,
or where the motion is such that any point will always lie on the
surface of a sphere of which the fixed j*fnt /is the center, as in
the universal and ball and socket joints. Such motion is called
spheric motion and is not nearly so common as the plane motion.
15. Screw Motion. A third class of motions occurs where a
body has a motion of rotation about an axis and also a motion
of translation along the axis at the same time, the motion of
translation bearing a fixed ratio to the motion of rotation. This
motion is called helical or screw motion and occurs quite
In the ordinary monkey wrench the movable jaw has a, plane
motion relative to the part held in the hand, the plane motion
being one of translation or sliding, all points on the screw have
plane motion relative to the part held, the motion being one of
rotation about the axis of the screw, and the screw has a helical
motion relative to the movable jaw, and vice versa.
PLANE CONSTRAINED MOTION
It has been noticed already that plane motion is frequently
constrained by causing a body to rotate about a given axis or by
causing the body to move along a straight line in a motion of
translation, the first form of motion may be called turning motion,
the latter form sliding motion.
16. Turning Motion. This may be constrained in many ways
and Fig. 1 shows several methods, where a shaft runs in a fixed
bearing, this shaft carrying a pulley as shown in the upper left
figure, while the lower left figure shows a thrust bearing for the
propeller shaft of a boat. In the figure (a), there is a pulley P
keyed to a straight shaft S which passes through a bearing B,
and if the construction were left in this form it would permit
plane turning motion in the pulley and shaft, but would not
constrain it, as the shaft might move axially through B. If,
however, two collars C are secured to the shaft by screws as
shown, then these collars effectually prevent the axial motion and
make only pure turning possible. On the propeller shaft at (6)
the collars C are forged on the shaft, a considerable number being
used on account of the great force tending to push the shaft
axially. Thus in both cases the relative turning motion is neces-
sitated by the two bodies, the shaft with its collars forming one
and the bearing the other, and these together are called a turning
pair for obvious reasons, the pair consisting of two elements.
It is evident that the turning pair may be arranged by other
constructions such as those shown on the right in Fig. 1, the form
used depending upon circumstances. The diagram (c) shows
in outline the method used in railroad cars, the bearing coming
in contact with the shaft only for a small part of the cir-
cumference of the latter, the two being held in contact purely
because of the connection to the car which rests on top of B,
and the collars C are here of slightly different form. At (d)
is a vertical bearing which, in a somewhat better form is often
used in turbines, but here again it is only possible to insure turn-
ing motion provided the weight is on the vertical shaft and
presses it into B. In this case there is only one part correspond-
ing to the collar C, which is the part of B below the shaft. At
(e) is a ball bearing used to support a car on top of a truck, the
weight of the car holding the balls in action.
17. Chain and Force Closure. In the cases (a) and (6), turn-
ing motion will take place by construction, and is said to be
secured by chain closure, which will be referred to later, while
in the cases (c), (d) and .(e) the motion is only constrained so long
as the external forces act in such a way as to press the two
elements of the pair together, plane motion being secured by
force closure. In cases, such as those described, where force
closure is permissible, it forms the cheaper construction, as a
18. Sliding Motion. The sliding pair also consists of two
elements, and if a section of these elements is taken normal to
the direction of sliding the elements must be non-circular. As
in the previous case the sliding pair in practice has very many
forms, a few of which are shown in Fig. 2, (a), (6), (c) and (d)
being forms in common use for the crossheads of steam engines,
(6) and (c) being rather cheaper in general than the others.
At (e), (/) and (
change gears and other similar places where there is little sliding;
(e) consists of a gear with a long keyway cut in it while the other
element has a parallel key, or "feather," fastened to it, so that
the outer element may slide along the shaft but cannot rotate
upon it. The construction of the forms (e) and (/) is evident.
The reader will see very many forms of this pair in machines and
should study them carefully.
In the automobile engine and in all the smaller gas and gasoline
engines, the sliding pair is circular, because the crosshead is
omitted and the connecting rod is directly attached to the piston,
the latter being circular and not constraining sliding motion.
In this case the sliding motion is constrained through the con-
necting rod, which on account of the pairing at its two ends
will not permit the piston to rotate. The real sliding pair, of
course, consists of the cylinder and piston, both of which are
circular, and constrainment is by force closure.
In the case of sliding pairs also it is possible to have chain
closure where constraint is due to the construction, as in the cases
illustrated in Fig. 2; in these cases the motion being one of
sliding irrespective of the directions of the acting forces. Fre-
quently, however, force closure is used as in the case (6) shown at
Fig. 3 which represents a planer table, the weight of which
alone keeps it in place. Occasionally through an accident the
planer table may be pushed out of place by a pressure on the
side, but, of course, the planer is not again used until the table is
replaced, for the reason that the design is such that the table
is only to have plane motion, a condition only possible if the
table rests in the grooves in the frame. In Fig. 3 (a) the same
table is constrained by chain closure and the tail sliding piece of
the piston rod in Fig. 3 (c) by force closure as is evident.
19. Lower and Higher Pairs. The two principal forms of
plane constrained motion are thus turning and sliding, these
motions being controlled by turning and sliding pairs respect-
ively, and each pair consisting of two elements. Where contact
between the two elements of a pair is over a surface the pair is
called a lower pair, and where the contact is only along a line
or at a point, the pair is called a higher pair. To illustrate this
the ordinary bearing may be taken as a very common example
of lower pairing, whereas a roller bearing has line contact and
a ball bearing point contact and are examples of higher pairing,
these illustrations are so familiar as to require no drawings. The
contact between spur gear teeth is along a line and therefore an
example of higher pairing.
In general, the lower pairs last longer than the higher, because
of the greater surface exposed for wear, but the conditions of the
problem settle the type of pairing. Thus, lower pairing is used on
the main shafts of large engines and turbines, but for automobiles
and bicycles the roller and ball bearings are common.
MACHINES, MECHANISMS, ETC.
20. Formation of Machines. Returning now to the steam
engine, Fig. 4, its formation may be further studied. The
valve gear and governor will be omitted at present and the
remaining parts discussed, these consist of the crank, crankshaft
and flywheel, the connecting rod, the piston, piston rod and
crosshead, and finally the frame and cylinder. Taking the
connecting rod b it is seen to contain two turning elements, one
at either end, and the real function of the metal in the rod is
to keep these two elements parallel and at a fixed distance apart.
The crank and crankshaft a contains two turning elements, one
of which is paired with one of the elements on the connecting rod
6, and forms the crankpin, and the other is paired with a corre-
sponding element on the frame d, forming the main bearing.
It is true that the main bearing may be made in two parts, both
of which are made on the frame, as in center-crank engines, or
one of which may be placed as an outboard bearing, but it will
readily be understood that this division of the bearing is only a
matter of practical convenience, for it is quite conceivable that
the bearing might be made in one piece, and if this piece were
long enough it would serve the purpose perfectly. Thus the
crank consists essentially of two turning elements properly
Again, the frame d contains the outer element of a turning
pair, of which the inner element is the crankshaft, and it also
contains a sliding element which is usually again divided into two
parts for the purpose of convenience in construction, the parts
being the crosshead guides and the cylinder. But the two parts
are not absolutely essential, for in the single-acting gasoline
engine, the guides are omitted and the sliding element is entirely
in the cylinder. Of course, the shape of the element depends
upon the purpose to which it is put; thus in the case last referred
to it is round.
Then, there is the crosshead c, with the turning element
pairing with the connecting rod and the sliding element pairing
with the sliding element on the frame. The sliding element
is usually in two parts to suit those of the frame, but it may be
only in one if so desired and conditions permit of it (see Fig. 2) .
Thus, the steam engine consists of four parts, each part con-
taining two elements of a pair, in some cases the elements being
for sliding, and in others for turning.
Again, on examining the small gasoline engine illustrated in
Fig. 5, it will be seen that the same method is adopted here as in
the steam engine, but the crosshead, piston and piston rod are all
combined in the single piston c. Further, in the Scotch yoke,
Fig. 6, a scheme in use for pumps of small sizes as well as on fire
engines of some makes and for other purposes, there is the
crank a with two turning elements, the piston and crosshead c
with two sliding elements, and the block 6, and the frame d,
each with one turning element and one sliding element.
21. Links and Chains. The same will be found true in all
machines having plane motion; each part contains at least two
elements, each of which is paired with corresponding elements
on the adjacent parts. For convenience each of these parts
of the machine is called a Unk, and the series of links so con-
nected as to give a complete machine is called a kinematic chain,
or simply a chain. It must be very carefully borne in mind that
if a kinematic chain is to form part of a machine or a whole
machine, then all the links must be so connected as to have definite
relative motions, this being an essential condition of the machine.
In Fig. 7 three cases are shown in which each link has two
turning elements. Case (a) could not form part of a machine be-
cause the three links could have no relative motion whatever, as
is evident by inspection, while at (6) it would be quite impossible
to move any link without the others having corresponding
changes of position, and for a given change in the relative posi-
tions of two of the links a definite change is produced in the
others. Looking next at case (c) , it is observed at once that -both
DC and OD could be secured to the ground and yet AB, BC, and
OA moved, that is a definite change in AB produces no necessary
change in OD or in CD, or one link may move without all the
others undergoing motion or relative change of position. Such
an arrangement could not form part of a machine because the
relative motions of the parts are not fixed but variable according
to conditions. At (d) is a chain which can be used, because if
any one link move relatively to any other, all the links move re-
latively, or if one link, say OD, is fastened to the ground and
OA moved, then must all the other links move.
22. Mechanisms. When a chain is used as a machine, usually
one of the links acts as the frame and is fixed to a foundation or
other stationary body. In studying the motions of various
links it is not necessary to know the exact shape of the links at all,
for the motion is completely known if the location and form of
the pairs of elements is known. Thus, the actual link may be
replaced by a straight bar which connects the elements of the
link together, and it will always be assumed that this bar never
changes its shape or length during motion. Thus, the chain
will be represented by straight lines and a chain so represented
having the relative motions of all links completely constrained
and having one link fixed will be called a mechanism.
23. Simple and Compound Chains. If the links of a chain
have only two elements each, the chain is said to be simple,
but if any link has three or more elements, as AB or BD, in Fig.
7 (d), the chain is compound.
24. Inversion of the Chain. Since in forming a mechanism
one link of the chain is fixed, it would appear that since any of
the links may be fixed in a given chain, it may be possible to
change the nature of the resulting mechanism by fixing various
links successively. Take as an example the mechanism shown
at (1) Fig. 8, d being the fixed link; here a would describe a circle,
c would swing about C and b would have a pendulum motion,
but with a moving pivot B. If b is fixed instead of d, a still
rotates, c swings about B and d now has the motion b originally
had, or the mechanism is unchanged.
If a is fixed then the whole mechanism may rotate, 6 and d
rotating about A and respectively as shown, and c also rotating,
the form of the mechanism being thus changed to one in which all
the links rotate. If, on the other hand, c is fixed, then none of
the links can rotate, but b and d simply oscillate about B and C
respectively. The reader will do well to make a cardboard model
to illustrate this point.
The process by which the nature of the mechanism is altered by
changing the fixed link is called inversion of the chain, and in
general, there are as many mechanisms as there are links in the
chain of which it is composed, although in the above illustra-
tion there are only three for the four links.
25. Slider-crank Chain. This inversion of the chain is very
well illustrated in case of the chain used in the steam engine,
which will be referred to in future as the slider-crank chain.
The mechanism is shown in Fig. 9 with the crank a, connecting
rod b and piston c, the latter having one sliding and one turning
element and representing the reciprocating masses, i.e., piston,
piston rod and crosshead. The frame d is represented by a
straight line and although it is common, yet the line of motion of c
does not always pass through 0; however, as shown at (1), it
represents the usual construction for the ordinary engine. If
now, instead of fixing d, b is fastened to the foundation, b being
the longer of the two links containing the two turning elements,
then a still rotates, c merely swings about Q and d has a swinging
and sliding motion, and -if c is a cylinder and a piston is attached
to d the result is the oscillating engine as shown at (2) Fig. 9,
and drawn in some detail in Fig. 10.
If instead of fixing the long rod b with the two turning elements,
the shorter rod a is fixed as shown at (3) , then b and d revolve
about P and respectively, and c also revolves sliding up and
down on d. If 6 is driven by means of a belt and pulley at
constant speed, then the angular velocity of d is variable and the
device may be used as a quick-return motion; in fact, it is em-
ployed in the Whitworth quick-return motion. The practical
form is also shown, Fig. 11, and the relation between the mechan-
ism and the actual machine will be readily discovered with the
help of the same letters.
In the Whitworth quick-return motion, Fig. 11, the pinion
is driven by belt and meshes with the gear 6. The gear rotates
on a large bearing E attached to the frame a of the machine, and
through the bearing E is a pin F, to one side of the center of E,
carrying the piece d, the latter being driven from 6 by a pin c
working in a slot in d. The arm A is attached to a tool holder
The Gnome motor used on aeroplanes is also an example of
this same inversion. It is shown in Fig. 12 and the cylinder
shown at the top with its rod and piston form the same mechanism
as the Whitworth quick-return motion, a being the link between
the shaft and lower connecting-rod centers. Study the mechanism
used with the other cylinders.
The fourth inversion found by fixing c is rarely used though it is
found occasionally. It is shown at (4) Fig. 9.
There are thus four inversions of this chain and it might be
further changed slightly by placing Q to one side of the link d
so that the/line of motion of Q, Fig. 9 (1), passses above 0, giving
the scheme used in operating the sleeves in some forms of gasoline
A somewhat different modification of the slider-crank chain is
shown at Fig. 13 a device also used -as a quick-return motion in
shapers and other machines. On comparing it with the Whit-
worth motion shown at Fig. 11, and the engine shown at Fig. 10,
it is seen that the mechanism of Fig. 9 may be somewhat altered
by varying the proportions of the links. The mechanism illus-
trated at Fig. 13 should be clear without further explanation.
D is the driving pinion working in with the large gear b, the tool
is attached to B which is driven from c by the link A. It is
readily seen that B moves faster in one direction than the other.
Further, an arrangement is made for varying the stroke of B
at pleasure by moving the center of c closer to, or further from,
that of 6.
26. Double Slider-crank Chain. A further illustration of a
chain which goes through many inversions in practice is given
in Fig. 14 and contains two links, b and d, with one sliding and
one turning element each, also one link a
with two turning elements and one c with
two sliding elements. When the link d
is fixed, c has a reciprocating motion and
such a setting is frequently used for small
pumps driven by belt through the crank
a (Fig. 14), c being the plunger. A detail
of this has already been given in Fig. 6.
With a fixed the device becomes Oldham's coupling which is
used to connect two parallel shafts nearly in line, Fig. 15. In
the figure b and d are two shafts which are parallel and rotate
about fixed axes. Keyed to each shaft is a half coupling with a
slot running across the center of its face and between these half
couplings is a peice c with two keys at right angles to each other,
one on each side, fitting in grooves in b and d. As b and d
revolve, c works sideways and vertically, both shafts always turn-
ing at the same speed. Points on c describe ellipses and a modi-
fication of the device has been used on elliptical chucks and on
instruments for drawing ellipses.
QUESTIONS ON CHAPTER I
1. Define the term machine and show that a gas engine, a stone crusher
and a planer are machines. Is a plough or a hay rake or hay fork a ma-
2. What are the methods of constrainment employed in the following:
Line shafting, loose pulley, sprocket chain, engine crankshaft, lathe spindle,
eccentric sheave, automobile clutch, change gear, belt. Which are by force
and which by chain closure?
3. Make a classification of the following with regard to constrainment
and the form of closure: Gas-engine piston, lathe carriage, milling-machine
head, ordinary D-slide valve, locomotive crosshead, valve rod, locomotive
link. Give a sketch to illustrate each. Why would force closure not do
for a connecting rod?
4. What form of pairing is used in the cases given in the above two ques-
tions? Is lower or higher pairing used in the following, and what is the
type of contact: Roller bearing, ball bearing, vertical-step bearing, cam and
roller in sewing machine, gear teeth, piston?
6. Define plane, helical and spherical motion. What form is used in the
parts above mentioned, and in a pair of bevel gears?
6. In helical motion if the pitch of the helix is zero, what form of motion
results; also what form for infinite pitch?
7. What is the resulting form of motion if the radius for a spherical
motion becomes infinitely great?
8. Show that all the motions in an ordinary engine but that of the gover-
nor balls are plane. What form of motion do the latter have?
9. Define and illustrate the following terms : Element, lower pair, higher
pair, link, chain, mechanism and compound chain.
10. List the links and their elements and give the form of motion and
method of constrainment in the parts of a locomotive side rod, beam engine,
stone crusher (Fig. 95) and shear (Fig. 94).
11. Explain and illustrate the inversion of the chain. Show that the
epicyclic gear train is an inversion of the ordinary train.
1 By a motion of translation is meant that all points on the part considered
move in parallel straight lines in the same direction and sense and through
the same distance.
Being adopted from
THE THEORY OF MACHINES , 1917
ROBERT W. ANGUS, B.A.Sc.,
MEMBER OF THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS,
PROFESSOR OF MECHANICAL ENGINEERING, UNIVERSITY OF
TORONTO, TORONTO, CANADA