Chapter II: On Applied Dynamics (1)

§ 83. _Laws of Motion._--The action of a machine in transmitting
_force_ and _motion_ simultaneously, or performing _work_, is
governed, in common with the phenomena of moving bodies in general, by
two "laws of motion."

_Division 1. Balanced Forces in Machines of Uniform Velocity._

§ 84. _Application of Force to Mechanism._--Forces are applied in
units of weight; and the unit most commonly employed in Britain is the
_pound avoirdupois_. The action of a force applied to a body is always
in reality distributed over some definite space, either a volume of
three dimensions or a surface of two. An example of a force
distributed throughout a volume is the _weight_ of the body itself,
which acts on every particle, however small. The _pressure_ exerted
between two bodies at their surface of contact, or between the two
parts of one body on either side of an ideal surface of separation, is
an example of a force distributed over a surface. The mode of
distribution of a force applied to a solid body requires to be
considered when its stiffness and strength are treated of; but, in
questions respecting the action of a force upon a rigid body
considered as a whole, the _resultant_ of the distributed force,
determined according to the principles of statics, and considered as
acting in a _single line_ and applied at a _single point_, may, for
the occasion, be substituted for the force as really distributed.
Thus, the weight of each separate piece in a machine is treated as
acting wholly at its _centre of gravity_, and each pressure applied to
it as acting at a point called the _centre of pressure_ of the surface
to which the pressure is really applied.

§ 85. _Forces applied to Mechanism Classed._--If [theta] be the
_obliquity_ of a force F applied to a piece of a machine--that is, the
angle made by the direction of the force with the direction of motion
of its point of application--then by the principles of statics, F may
be resolved into two rectangular components, viz.:--

Along the direction of motion, P = F cos [theta] \ (49)
Across the direction of motion, Q = F sin [theta] /

If the component along the direction of motion acts with the motion,
it is called an _effort_; if _against_ the motion, a _resistance_. The
component _across_ the direction of motion is a _lateral pressure_;
the unbalanced lateral pressure on any piece, or part of a piece, is
_deflecting force_. A lateral pressure may increase resistance by
causing friction; the friction so caused acts against the motion, and
is a resistance, but the lateral pressure causing it is not a
resistance. Resistances are distinguished into _useful_ and
_prejudicial_, according as they arise from the useful effect produced
by the machine or from other causes.

§ 86. _Work._--_Work_ consists in moving against resistance. The work
is said to be _performed_, and the resistance _overcome_. Work is
measured by the product of the resistance into the distance through
which its point of application is moved. The _unit of work_ commonly
used in Britain is a resistance of one pound overcome through a
distance of one foot, and is called a _foot-pound_.

Work is distinguished into _useful work_ and _prejudicial_ or _lost
work_, according as it is performed in producing the useful effect of
the machine, or in overcoming prejudicial resistance.

§ 87. _Energy: Potential Energy._--_Energy_ means _capacity for
performing work_. The _energy of an effort_, or _potential energy_, is
measured by the product of the effort into the distance through which
its point of application is _capable_ of being moved. The unit of
energy is the same with the unit of work.

When the point of application of an effort _has been moved_ through a
given distance, energy is said to have been _exerted_ to an amount
expressed by the product of the effort into the distance through which
its point of application has been moved.

§ 88. _Variable Effort and Resistance._--If an effort has different
magnitudes during different portions of the motion of its point of
application through a given distance, let each different magnitude of
the effort P be multiplied by the length [Delta]s of the corresponding
portion of the path of the point of application; the sum

[Sigma] · P[Delta]s (50)

is the whole energy exerted. If the effort varies by insensible
gradations, the energy exerted is the integral or limit towards which
that sum approaches continually as the divisions of the path are made
smaller and more numerous, and is expressed by

[int]P ds. (51)

Similar processes are applicable to the finding of the work performed
in overcoming a varying resistance.

The work done by a machine can be actually measured by means of a
dynamometer (q.v.).

§ 89. _Principle of the Equality of Energy and Work._--From the first
law of motion it follows that in a machine whose pieces move with
uniform velocities the efforts and resistances must balance each
other. Now from the laws of statics it is known that, in order that a
system of forces applied to a system of connected points may be in
equilibrium, it is necessary that the sum formed by putting together
the products of the forces by the respective distances through which
their points of application are capable of moving simultaneously, each
along the direction of the force applied to it, shall be
zero,--products being considered positive or negative according as the
direction of the forces and the possible motions of their points of
application are the same or opposite.

In other words, the sum of the negative products is equal to the sum
of the positive products. This principle, applied to a machine whose
parts move with uniform velocities, is equivalent to saying that in
any given interval of time _the energy exerted is equal to the work
performed_.

The symbolical expression of this law is as follows: let efforts be
applied to one or any number of points of a machine; let any one of
these efforts be represented by P, and the distance traversed by its
point of application in a given interval of time by ds; let
resistances be overcome at one or any number of points of the same
machine; let any one of these resistances be denoted by R, and the
distance traversed by its point of application in the given interval
of time by ds´; then

[Sigma] · P ds = [Sigma] · R ds´. (52)

The lengths ds, ds´ are proportional to the velocities of the points
to whose paths they belong, and the proportions of those velocities to
each other are deducible from the construction of the machine by the
principles of pure mechanism explained in Chapter I.

§ 90. _Static Equilibrium of Mechanisms._--The principle stated in the
preceding section, namely, that the energy exerted is equal to the
work performed, enables the ratio of the components of the forces
acting in the respective directions of motion at two points of a
mechanism, one being the point of application of the effort, and the
other the point of application of the resistance, to be readily found.
Removing the summation signs in equation (52) in order to restrict its
application to two points and dividing by the common time interval
during which the respective small displacements ds and ds´ were made,
it becomes P ds/dt = R ds´/dt, that is, Pv = Rv´, which shows that the
force ratio is the inverse of the velocity ratio. It follows at once
that any method which may be available for the determination of the
velocity ratio is equally available for the determination of the force
ratio, it being clearly understood that the forces involved are the
components of the actual forces resolved in the direction of motion
of the points. The relation between the effort and the resistance may
be found by means of this principle for all kinds of mechanisms, when
the friction produced by the components of the forces across the
direction of motion of the two points is neglected. Consider the
following example:--

A four-bar chain having the configuration shown in fig. 126 supports a
load P at the point x. What load is required at the point y to
maintain the configuration shown, both loads being supposed to act
vertically? Find the instantaneous centre O_(bd), and resolve each
load in the respective directions of motion of the points x and y;
thus there are obtained the components P cos [theta] and R cos [phi].
Let the mechanism have a small motion; then, for the instant, the link
b is turning about its instantaneous centre O_(bd), and, if [omega] is
its instantaneous angular velocity, the velocity of the point x is
[omega]r, and the velocity of the point y is [omega]s. Hence, by the
principle just stated, P cos [theta] × [omega]r = R cos [phi] ×
[omega]s. But, p and q being respectively the perpendiculars to the
lines of action of the forces, this equation reduces to P_p = R_q,
which shows that the ratio of the two forces may be found by taking
moments about the instantaneous centre of the link on which they act.

The forces P and R may, however, act on different links. The general
problem may then be thus stated: Given a mechanism of which r is the
fixed link, and s and t any other two links, given also a force f_s,
acting on the link s, to find the force f_t acting in a given
direction on the link t, which will keep the mechanism in static
equilibrium. The graphic solution of this problem may be effected
thus:--

(1) Find the three virtual centres O_(rs), O_(rt), O_(st), which
must be three points in a line.

(2) Resolve f_s into two components, one of which, namely, f_q,
passes through O_(rs) and may be neglected, and the other f_p passes
through O_(st).

(3) Find the point M, where f_p joins the given direction of f_t,
and resolve f_p into two components, of which one is in the
direction MO_(rt), and may be neglected because it passes through
O_(rt), and the other is in the given direction of f_t and is
therefore the force required.

This statement of the problem and the solution is due to Sir A. B. W.
Kennedy, and is given in ch. 8 of his _Mechanics of Machinery_.
Another general solution of the problem is given in the _Proc. Lond.
Math. Soc._ (1878-1879), by the same author. An example of the method
of solution stated above, and taken from the _Mechanics of Machinery_,
is illustrated by the mechanism fig. 127, which is an epicyclic train
of three wheels with the first wheel r fixed. Let it be required to
find the vertical force which must act at the pitch radius of the last
wheel t to balance exactly a force f_s acting vertically downwards on
the arm at the point indicated in the figure. The two links concerned
are the last wheel t and the arm s, the wheel r being the fixed link
of the mechanism. The virtual centres O_(rs), O_(st) are at the
respective axes of the wheels r and t, and the centre O_(rt) divides
the line through these two points externally in the ratio of the train
of wheels. The figure sufficiently indicates the various steps of the
solution.

The relation between the effort and the resistance in a machine to
include the effect of friction at the joints has been investigated in
a paper by Professor Fleeming Jenkin, "On the application of graphic
methods to the determination of the efficiency of machinery" (_Trans.
Roy. Soc. Ed._, vol. 28). It is shown that a machine may at any
instant be represented by a frame of links the stresses in which are
identical with the pressures at the joints of the mechanism. This
self-strained frame is called the _dynamic frame_ of the machine. The
driving and resisting efforts are represented by elastic links in the
dynamic frame, and when the frame with its elastic links is drawn the
stresses in the several members of it may be determined by means of
reciprocal figures. Incidentally the method gives the pressures at
every joint of the mechanism.

§ 91. _Efficiency._--The _efficiency_ of a machine is the ratio of the
_useful_ work to the _total_ work--that is, to the energy exerted--and
is represented by

[Sigma]·R_u ds´ [Sigma]·R_u ds´ [Sigma]·R_u ds´ U
--------------- = --------------------------------- = --------------- = ---. (53)
[Sigma]·R ds´ [Sigma]·R_u ds´ + [Sigma]·R_p ds´ [Sigma]·P ds E

R_u being taken to represent useful and R_p prejudicial resistances.
The more nearly the efficiency of a machine approaches to unity the
better is the machine.

§ 92. _Power and Effect._--The _power_ of a machine is the energy
exerted, and the _effect_ the useful work performed, in some interval
of time of definite length, such as a second, an hour, or a day.

The unit of power, called conventionally a horse-power, is 550
foot-pounds per second, or 33,000 foot-pounds per minute, or 1,980,000
foot-pounds per hour.

§ 93. _Modulus of a Machine._--In the investigation of the properties
of a machine, the useful resistances to be overcome and the useful
work to be performed are usually given. The prejudicial resistances
arc generally functions of the useful resistances of the weights of
the pieces of the mechanism, and of their form and arrangement; and,
having been determined, they serve for the computation of the _lost_
work, which, being added to the useful work, gives the expenditure of
energy required. The result of this investigation, expressed in the
form of an equation between this energy and the useful work, is called
by Moseley the _modulus_ of the machine. The general form of the
modulus may be expressed thus--

E = U + [phi](U, A) + [psi](A), (54)

where A denotes some quantity or set of quantities depending on the
form, arrangement, weight and other properties of the mechanism.
Moseley, however, has pointed out that in most cases this equation
takes the much more simple form of

E = (1 + A)U + B, (55)

where A and B are _constants_, depending on the form, arrangement and
weight of the mechanism. The efficiency corresponding to the last
equation is

U 1
--- = -----------. (56)
E 1 + A + B/U

§ 94. _Trains of Mechanism._--In applying the preceding principles to
a train of mechanism, it may either be treated as a whole, or it may
be considered in sections consisting of single pieces, or of any
convenient portion of the train--each section being treated as a
machine, driven by the effort applied to it and energy exerted upon it
through its line of connexion with the preceding section, performing
useful work by driving the following section, and losing work by
overcoming its own prejudicial resistances. It is evident that _the
efficiency of the whole train is the product of the efficiencies of
its sections_.

§ 95. _Rotating Pieces: Couples of Forces._--It is often convenient to
express the energy exerted upon and the work performed by a turning
piece in a machine in terms of the _moment_ of the _couples of forces_
acting on it, and of the angular velocity. The ordinary British unit
of moment is a _foot-pound_; but it is to be remembered that this is a
foot-pound of a different sort from the unit of energy and work.

If a force be applied to a turning piece in a line not passing through
its axis, the axis will press against its bearings with an equal and
parallel force, and the equal and opposite reaction of the bearings
will constitute, together with the first-mentioned force, a couple
whose arm is the perpendicular distance from the axis to the line of
action of the first force.

A couple is said to be _right_ or _left handed_ with reference to the
observer, according to the direction in which it tends to turn the
body, and is a _driving_ couple or a _resisting_ couple according as
its tendency is with or against that of the actual rotation.

Let dt be an interval of time, [alpha] the angular velocity of the
piece; then [alpha]dt is the angle through which it turns in the
interval dt, and ds = vdt = r[alpha]dt is the distance through which
the point of application of the force moves. Let P represent an
effort, so that Pr is a driving couple, then

P ds = Pv dt = Pr[alpha] dt = M[alpha] dt (57)

is the energy exerted by the couple M in the interval dt; and a
similar equation gives the work performed in overcoming a resisting
couple. When several couples act on one piece, the resultant of their
moments is to be multiplied by the common angular velocity of the
whole piece.

§ 96. _Reduction of Forces to a given Point, and of Couples to the
Axis of a given Piece._--In computations respecting machines it is
often convenient to substitute for a force applied to a given point,
or a couple applied to a given piece, the _equivalent_ force or couple
applied to some other point or piece; that is to say, the force or
couple, which, if applied to the other point or piece, would exert
equal energy or employ equal work. The principles of this reduction
are that the ratio of the given to the equivalent force is the
reciprocal of the ratio of the velocities of their points of
application, and the ratio of the given to the equivalent couple is
the reciprocal of the ratio of the angular velocities of the pieces to
which they are applied.

These velocity ratios are known by the construction of the mechanism,
and are independent of the absolute speed.

§ 97. _Balanced Lateral Pressure of Guides and Bearings._--The most
important part of the lateral pressure on a piece of mechanism is the
reaction of its guides, if it is a sliding piece, or of the bearings
of its axis, if it is a turning piece; and the balanced portion of
this reaction is equal and opposite to the resultant of all the other
forces applied to the piece, its own weight included. There may be or
may not be an unbalanced component in this pressure, due to the
deviated motion. Its laws will be considered in the sequel.

§ 98. _Friction. Unguents._--The most important kind of resistance in
machines is the _friction_ or _rubbing resistance_ of surfaces which
slide over each other. The _direction_ of the resistance of friction
is opposite to that in which the sliding takes place. Its _magnitude_
is the product of the _normal pressure_ or force which presses the
rubbing surfaces together in a direction perpendicular to themselves
into a specific constant already mentioned in § 14, as the
_coefficient of friction_, which depends on the nature and condition
of the surfaces of the unguent, if any, with which they are covered.
The _total pressure_ exerted between the rubbing surfaces is the
resultant of the normal pressure and of the friction, and its
_obliquity_, or inclination to the common perpendicular of the
surfaces, is the _angle of repose_ formerly mentioned in § 14, whose
tangent is the coefficient of friction. Thus, let N be the normal
pressure, R the friction, T the total pressure, f the coefficient of
friction, and [phi] the angle of repose; then

f = tan [phi] \ (58)
R = fN = N tan [phi] = T sin [phi] /

Experiments on friction have been made by Coulomb, Samuel Vince, John
Rennie, James Wood, D. Rankine and others. The most complete and
elaborate experiments are those of Morin, published in his _Notions
fondamentales de mécanique_, and republished in Britain in the works
of Moseley and Gordon.

The experiments of Beauchamp Tower ("Report of Friction Experiments,"
_Proc. Inst. Mech. Eng._, 1883) showed that when oil is supplied to a
journal by means of an oil bath the coefficient of friction varies
nearly inversely as the load on the bearing, thus making the product
of the load on the bearing and the coefficient of friction a constant.
Mr Tower's experiments were carried out at nearly constant
temperature. The more recent experiments of Lasche (_Zeitsch, Verein
Deutsche Ingen._, 1902, 46, 1881) show that the product of the
coefficient of friction, the load on the bearing, and the temperature
is approximately constant. For further information on this point and
on Osborne Reynolds's theory of lubrication see BEARINGS and
LUBRICATION.

§ 99. _Work of Friction. Moment of Friction._--The work performed in a
unit of time in overcoming the friction of a pair of surfaces is the
product of the friction by the velocity of sliding of the surfaces
over each other, if that is the same throughout the whole extent of
the rubbing surfaces. If that velocity is different for different
portions of the rubbing surfaces, the velocity of each portion is to
be multiplied by the friction of that portion, and the results summed
or integrated.

When the relative motion of the rubbing surfaces is one of rotation,
the work of friction in a unit of time, for a portion of the rubbing
surfaces at a given distance from the axis of rotation, may be found
by multiplying together the friction of that portion, its distance
from the axis, and the angular velocity. The product of the force of
friction by the distance at which it acts from the axis of rotation is
called the _moment of friction_. The total moment of friction of a
pair of rotating rubbing surfaces is the sum or integral of the
moments of friction of their several portions.

To express this symbolically, let du represent the area of a portion
of a pair of rubbing surfaces at a distance r from the axis of their
relative rotation; p the intensity of the normal pressure at du per
unit of area; and f the coefficient of friction. Then the moment of
friction of du is fprdu;

the total moment of friction is f [integral] pr·du; \
and the work performed in a unit cf time in overcoming friction, > (59)
when the angular velocity is [alpha], is [alpha]f [int] pr·du. /

It is evident that the moment of friction, and the work lost by being
performed in overcoming friction, are less in a rotating piece as the
bearings are of smaller radius. But a limit is put to the diminution
of the radii of journals and pivots by the conditions of durability
and of proper lubrication, and also by conditions of strength and
stiffness.

§ 100. _Total Pressure between Journal and Bearing._--A single piece
rotating with a uniform velocity has four mutually balanced forces
applied to it: (l) the effort exerted on it by the piece which drives
it; (2) the resistance of the piece which follows it--which may be
considered for the purposes of the present question as useful
resistance; (3) its weight; and (4) the reaction of its own
cylindrical bearings. There are given the following data:--

The direction of the effort.
The direction of the useful resistance.
The weight of the piece and the direction in which it acts.
The magnitude of the useful resistance.
The radius of the bearing r.
The angle of repose [phi], corresponding to the friction of the
journal on the bearing.

And there are required the following:--

The direction of the reaction of the bearing.
The magnitude of that reaction.
The magnitude of the effort.

Let the useful resistance and the weight of the piece be compounded by
the principles of statics into one force, and let this be called _the
given force_.

The directions of the effort and of the given force are either
parallel or meet in a point. If they are parallel, the direction of
the reaction of the bearing is also parallel to them; if they meet in
a point, the direction of the reaction traverses the same point.

Also, let AAA, fig. 128, be a section of the bearing, and C its axis;
then the direction of the reaction, at the point where it intersects
the circle AAA, must make the angle [phi] with the radius of that
circle; that is to say, it must be a line such as PT touching the
smaller circle BB, whose radius is r · sin [phi]. The side on which it
touches that circle is determined by the fact that the obliquity of
the reaction is such as to oppose the rotation.

Thus is determined the direction of the reaction of the bearing; and
the magnitude of that reaction and of the effort are then found by the
principles of the equilibrium of three forces already stated in § 7.

The work lost in overcoming the friction of the bearing is the same as
that which would be performed in overcoming at the circumference of
the small circle BB a resistance equal to the whole pressure between
the journal and bearing.

In order to diminish that pressure to the smallest possible amount,
the effort, and the resultant of the useful resistance, and the weight
of the piece (called above the "given force") ought to be opposed to
each other as directly as is practicable consistently with the
purposes of the machine.

An investigation of the forces acting on a bearing and journal
lubricated by an oil bath will be found in a paper by Osborne Reynolds
in the _Phil. Trans._ pt. i. (1886). (See also BEARINGS.)

§ 101. _Friction of Pivots and Collars._--When a shaft is acted upon
by a force tending to shift it lengthways, that force must be balanced
by the reaction of a bearing against a _pivot_ at the end of the
shaft; or, if that be impossible, against one or more _collars_, or
rings _projecting_ from the body of the shaft. The bearing of the
pivot is called a _step_ or _footstep_. Pivots require great hardness,
and are usually made of steel. The _flat_ pivot is a cylinder of steel
having a plane circular end as a rubbing surface. Let N be the total
pressure sustained by a flat pivot of the radius r; if that pressure
be uniformly distributed, which is the case when the rubbing surfaces
of the pivot and its step are both true planes, the _intensity_ of the
pressure is

p = N/[pi]r²; (60)

and, introducing this value into equation 59, the _moment of friction
of the flat pivot_ is found to be

(2/3)fNr (61)

or two-thirds of that of a cylindrical journal of the same radius
under the same normal pressure.

The friction of a _conical_ pivot exceeds that of a flat pivot of the
same radius, and under the same pressure, in the proportion of the
side of the cone to the radius of its base.

The moment of friction of a _collar_ is given by the formula--

r³ - r´³
(2/3)fN --------, (62)
r² - r´²

where r is the external and r´ the internal radius.

In the _cup and ball_ pivot the end of the shaft and the step present
two recesses facing each other, into which art fitted two shallow cups
of steel or hard bronze. Between the concave spherical surfaces of
those cups is placed a steel ball, being either a complete sphere or a
lens having convex surfaces of a somewhat less radius than the concave
surfaces of the cups. The moment of friction of this pivot is at first
almost inappreciable from the extreme smallness of the radius of the
circles of contact of the ball and cups, but, as they wear, that
radius and the moment of friction increase.

It appears that the rapidity with which a rubbing surface wears away
is proportional to the friction and to the velocity jointly, or nearly
so. Hence the pivots already mentioned wear unequally at different
points, and tend to alter their figures. Schiele has invented a pivot
which preserves its original figure by wearing equally at all points
in a direction parallel to its axis. The following are the principles
on which this equality of wear depends:--

The rapidity of wear of a surface measured in an _oblique_ direction
is to the rapidity of wear measured normally as the secant of the
obliquity is to unity. Let OX (fig. 129) be the axis of a pivot, and
let RPC be a portion of a curve such that at any point P the secant of
the obliquity to the normal of the curve of a line parallel to the
axis is inversely proportional to the ordinate PY, to which the
velocity of P is proportional. The rotation of that curve round OX
will generate the form of pivot required. Now let PT be a tangent to
the curve at P, cutting OX in T; PT = PY × _secant obliquity_, and
this is to be a constant quantity; hence the curve is that known as
the _tractory_ of the straight line OX, in which PT = OR = constant.
This curve is described by having a fixed straight edge parallel to
OX, along which slides a slider carrying a pin whose centre is T. On
that pin turns an arm, carrying at a point P a tracing-point, pencil
or pen. Should the pen have a nib of two jaws, like those of an
ordinary drawing-pen, the plane of the jaws must pass through PT.
Then, while T is slid along the axis from O towards X, P will be drawn
after it from R towards C along the tractory. This curve, being an
asymptote to its axis, is capable of being indefinitely prolonged
towards X; but in designing pivots it should stop before the angle PTY
becomes less than the angle of repose of the rubbing surfaces,
otherwise the pivot will be liable to stick in its bearing. The moment
of friction of "Schiele's anti-friction pivot," as it is called, is
equal to that of a cylindrical journal of the radius OR = PT the
constant tangent, under the same pressure.

Records of experiments on the friction of a pivot bearing will be
found in the _Proc. Inst. Mech. Eng._ (1891), and on the friction of a
collar bearing ib. May 1888.

§ 102. _Friction of Teeth._--Let N be the normal pressure exerted
between a pair of teeth of a pair of wheels; s the total distance
through which they slide upon each other; n the number of pairs of
teeth which pass the plane of axis in a unit of time; then

nfNs (63)

is the work lost in unity of time by the friction of the teeth. The
sliding s is composed of two parts, which take place during the
approach and recess respectively. Let those be denoted by s1 and s2,
so that s = s1 + s2. In § 45 the _velocity_ of sliding at any instant
has been given, viz. u = c ([alpha]1 + [alpha]2), where u is that
velocity, c the distance T1 at any instant from the point of contact
of the teeth to the pitch-point, and [alpha]1, [alpha]2 the respective
angular velocities of the wheels.

Let v be the common velocity of the two pitch-circles, r1, r2, their
radii; then the above equation becomes

/ 1 1 \
u = cv ( --- + --- ).
\r1 r2 /

To apply this to involute teeth, let c1 be the length of the approach,
c2 that of the recess, u1, the _mean_ volocity of sliding during the
approach, u2 that during the recess; then

c1v / 1 1 \ c2v / 1 1 \
u1 = --- ( --- + --- ); u2 = --- ( --- + --- )
2 \r1 r2 / 2 \r1 r2 /

also, let [theta] be the obliquity of the action; then the times
occupied by the approach and recess are respectively

c1 c2
-------------, -------------;
v cos [theta] v cos [theta]

giving, finally, for the length of sliding between each pair of teeth,

c1² + c2² / 1 1 \
s = s1 + s2 = ------------- ( --- + --- ) (64)
2 cos [theta] \r1 r2 /

which, substituted in equation (63), gives the work lost in a unit of
time by the friction of involute teeth. This result, which is exact
for involute teeth, is approximately true for teeth of any figure.

For inside gearing, if r1 be the less radius and r2 the greater, 1/r1
- 1/r2 is to be substituted for 1/r1 + 1/r2.

§ 103. _Friction of Cords and Belts._--A flexible band, such as a
cord, rope, belt or strap, may be used either to exert an effort or a
resistance upon a pulley round which it wraps. In either case the
tangential force, whether effort or resistance, exerted between the
band and the pulley is their mutual friction, caused by and
proportional to the normal pressure between them.

Let T1 be the tension of the free part of the band at that side
_towards_ which it tends to draw the pulley, or _from_ which the
pulley tends to draw it; T2 the tension of the free part at the other
side; T the tension of the band at any intermediate point of its arc
of contact with the pulley; [theta] the ratio of the length of that
arc to the radius of the pulley; d[theta] the ratio of an indefinitely
small element of that arc to the radius; F = T1 - T2 the total
friction between the band and the pulley; dF the elementary portion of
that friction due to the elementary arc d[theta]; f the coefficient of
friction between the materials of the band and pulley.

Then, according to a well-known principle in statics, the normal
pressure at the elementary arc d[theta] is Td[theta], T being the mean
tension of the band at that elementary arc; consequently the friction
on that arc is dF = fTd[theta]. Now that friction is also the
difference between the tensions of the band at the two ends of the
elementary arc, or dT = dF = fTd[theta]; which equation, being
integrated throughout the entire arc of contact, gives the following
formulae:--

T1 \
hyp log. -- = f^[theta] |
T2 |
|
T1 > (65)
-- = ef^[theta] |
T2 |
|
F = T1 - T2 = T1(1 - e - f^[theta]) = T2(ef^[theta] - 1) /

When a belt connecting a pair of pulleys has the tensions of its two
sides originally equal, the pulleys being at rest, and when the
pulleys are next set in motion, so that one of them drives the other
by means of the belt, it is found that the advancing side of the belt
is exactly as much tightened as the returning side is slackened, so
that the _mean_ tension remains unchanged. Its value is given by this
formula--

T1 + T2 ef^[theta] + 1
------- = ----------------- (66)
2 2(ef^[theta] - 1)

which is useful in determining the original tension required to enable
a belt to transmit a given force between two pulleys.

The equations 65 and 66 are applicable to a kind of _brake_ called a
_friction-strap_, used to stop or moderate the velocity of machines by
being tightened round a pulley. The strap is usually of iron, and the
pulley of hard wood.

Let [alpha] denote the arc of contact expressed in _turns and
fractions of a turn_; then

[theta] = 6.2832a \ (67)
ef^[theta] = number whose common logarithm is 2.7288fa /

See also DYNAMOMETER for illustrations of the use of what are
essentially friction-straps of different forms for the measurement of
the brake horse-power of an engine or motor.

§ 104. _Stiffness of Ropes._--Ropes offer a resistance to being bent,
and, when bent, to being straightened again, which arises from the
mutual friction of their fibres. It increases with the sectional area
of the rope, and is inversely proportional to the radius of the curve
into which it is bent.

The _work lost_ in pulling a given length of rope over a pulley is
found by multiplying the length of the rope in feet by its stiffness
in pounds, that stiffness being the excess of the tension at the
leading side of the rope above that at the following side, which is
necessary to bend it into a curve fitting the pulley, and then to
straighten it again.

The following empirical formulae for the stiffness of hempen ropes
have been deduced by Morin from the experiments of Coulomb:--

Let F be the stiffness in pounds avoirdupois; d the diameter of the
rope in inches, n = 48d² for white ropes and 35d² for tarred ropes; r
the _effective_ radius of the pulley in inches; T the tension in
pounds. Then

n \
For white ropes, F = --- (0.0012 + 0.001026n + 0.0012T) |
r |
> (68)
n |
For tarred ropes, F = --- (0.006 + 0.001392n + 0.00168T) |
r /

§ 105. _Friction-Couplings._--Friction is useful as a means of
communicating motion where sudden changes either of force or velocity
take place, because, being limited in amount, it may be so adjusted as
to limit the forces which strain the pieces of the mechanism within
the bounds of safety. Amongst contrivances for effecting this object
are _friction-cones_. A rotating shaft carries upon a cylindrical
portion of its figure a wheel or pulley turning loosely on it, and
consequently capable of remaining at rest when the shaft is in motion.
This pulley has fixed to one side, and concentric with it, a short
frustum of a hollow cone. At a small distance from the pulley the
shaft carries a short frustum of a solid cone accurately turned to fit
the hollow cone. This frustum is made always to turn along with the
shaft by being fitted on a square portion of it, or by means of a rib
and groove, or otherwise, but is capable of a slight longitudinal
motion, so as to be pressed into, or withdrawn from, the hollow cone
by means of a lever. When the cones are pressed together or engaged,
their friction causes the pulley to rotate along with the shaft; when
they are disengaged, the pulley is free to stand still. The angle made
by the sides of the cones with the axis should not be less than the
angle of repose. In the _friction-clutch_, a pulley loose on a shaft
has a hoop or gland made to embrace it more or less tightly by means
of a screw; this hoop has short projecting arms or ears. A fork or
_clutch_ rotates along with the shaft, and is capable of being moved
longitudinally by a handle. When the clutch is moved towards the hoop,
its arms catch those of the hoop, and cause the hoop to rotate and to
communicate its rotation to the pulley by friction. There are many
other contrivances of the same class, but the two just mentioned may
serve for examples.

§ 106. _Heat of Friction: Unguents._--The work lost in friction is
employed in producing heat. This fact is very obvious, and has been
known from a remote period; but the _exact_ determination of the
proportion of the work lost to the heat produced, and the experimental
proof that that proportion is the same under all circumstances and
with all materials, solid, liquid and gaseous, are comparatively
recent achievements of J. P. Joule. The quantity of work which
produces a British unit of heat (or so much heat as elevates the
temperature of one pound of pure water, at or near ordinary
atmospheric temperatures, by 1° F.) is 772 foot-pounds. This constant,
now designated as "Joule's equivalent," is the principal experimental
datum of the science of thermodynamics.

A more recent determination (_Phil. Trans._, 1897), by Osborne
Reynolds and W. M. Moorby, gives 778 as the mean value of Joule's
equivalent through the range of 32° to 212° F. See also the papers of
Rowland in the _Proc. Amer. Acad._ (1879), and Griffiths, _Phil.
Trans._ (1893).

The heat produced by friction, when moderate in amount, is useful in
softening and liquefying thick unguents; but when excessive it is
prejudicial, by decomposing the unguents, and sometimes even by
softening the metal of the bearings, and raising their temperature so
high as to set fire to neighbouring combustible matters.

Excessive heating is prevented by a constant and copious supply of a
good unguent. The elevation of temperature produced by the friction of
a journal is sometimes used as an experimental test of the quality of
unguents. For modern methods of forced lubrication see BEARINGS.

§ 107. _Rolling Resistance._--By the rolling of two surfaces over each
other without sliding a resistance is caused which is called sometimes
"rolling friction," but more correctly _rolling resistance_. It is of
the nature of a _couple_, resisting rotation. Its _moment_ is found by
multiplying the normal pressure between the rolling surfaces by an
_arm_, whose length depends on the nature of the rolling surfaces, and
the work lost in a unit of time in overcoming it is the product of its
moment by the _angular velocity_ of the rolling surfaces relatively to
each other. The following are approximate values of the arm in
decimals of a foot:--

Oak upon oak 0.006 (Coulomb).
Lignum vitae on oak 0.004 "
Cast iron on cast iron 0.002 (Tredgold).

§ 108. _Reciprocating Forces: Stored and Restored Energy._--When a
force acts on a machine alternately as an effort and as a resistance,
it may be called a _reciprocating force_. Of this kind is the weight
of any piece in the mechanism whose centre of gravity alternately
rises and falls; for during the rise of the centre of gravity that
weight acts as a resistance, and energy is employed in lifting it to
an amount expressed by the product of the weight into the vertical
height of its rise; and during the fall of the centre of gravity the
weight acts as an effort, and exerts in assisting to perform the work
of the machine an amount of energy exactly equal to that which had
previously been employed in lifting it. Thus that amount of energy is
not lost, but has its operation deferred; and it is said to be
_stored_ when the weight is lifted, and _restored_ when it falls.

In a machine of which each piece is to move with a uniform velocity,
if the effort and the resistance be constant, the weight of each piece
must be balanced on its axis, so that it may produce lateral pressure
only, and not act as a reciprocating force. But if the effort and the
resistance be alternately in excess, the uniformity of speed may still
be preserved by so adjusting some moving weight in the mechanism that
when the effort is in excess it may be lifted, and so balance and
employ the excess of effort, and that when the resistance is in excess
it may fall, and so balance and overcome the excess of
resistance--thus _storing_ the periodical excess of energy and
_restoring_ that energy to perform the periodical excess of work.

Other forces besides gravity may be used as reciprocating forces for
storing and restoring energy--for example, the elasticity of a spring
or of a mass of air.

In most of the delusive machines commonly called "perpetual motions,"
of which so many are patented in each year, and which are expected by
their inventors to perform work without receiving energy, the
fundamental fallacy consists in an expectation that some reciprocating
force shall restore more energy than it has been the means of storing.

_Division 2. Deflecting Forces._

§ 109. _Deflecting Force for Translation in a Curved Path._--In
machinery, deflecting force is supplied by the tenacity of some piece,
such as a crank, which guides the deflected body in its curved path,
and is _unbalanced_, being employed in producing deflexion, and not in
balancing another force.

§ 110. _Centrifugal Force of a Rotating Body._--_The centrifugal force
exerted by a rotating body on its axis of rotation is the same in
magnitude as if the mass of the body were concentrated at its centre
of gravity, and acts in a plane passing through the axis of rotation
and the centre of gravity of the body._

The particles of a rotating body exert centrifugal forces on each
other, which strain the body, and tend to tear it asunder, but these
forces balance each other, and do not affect the resultant centrifugal
force exerted on the axis of rotation.[3]

_If the axis of rotation traverses the centre of gravity of the body,
the centrifugal force exerted on that axis is nothing._

Hence, unless there be some reason to the contrary, each piece of a
machine should be balanced on its axis of rotation; otherwise the
centrifugal force will cause strains, vibration and increased
friction, and a tendency of the shafts to jump out of their bearings.

§ 111. _Centrifugal Couples of a Rotating Body._--Besides the tendency
(if any) of the combined centrifugal forces of the particles of a
rotating body to _shift_ the axis of rotation, they may also tend to
_turn_ it out of its original direction. The latter tendency is called
_a centrifugal couple_, and vanishes for rotation about a principal
axis.

It is essential to the steady motion of every rapidly rotating piece
in a machine that its axis of rotation should not merely traverse its
centre of gravity, but should be a permanent axis; for otherwise the
centrifugal couples will increase friction, produce oscillation of the
shaft and tend to make it leave its bearings.

The principles of this and the preceding section are those which
regulate the adjustment of the weight and position of the
counterpoises which are placed between the spokes of the
driving-wheels of locomotive engines.

FIG. 130.]

§ 112.* _Method of computing the position and magnitudes of balance
weights which must be added to a given system of arbitrarily chosen
rotating masses in order to make the common axis of rotation a
permanent axis._--The method here briefly explained is taken from a
paper by W. E. Dalby, "The Balancing of Engines with special reference
to Marine Work," _Trans. Inst. Nav. Arch._ (1899). Let the weight
(fig. 130), attached to a truly turned disk, be rotated by the shaft
OX, and conceive that the shaft is held in a bearing at one point, O.
The force required to constrain the weight to move in a circle, that
is the deviating force, produces an equal and opposite reaction on the
shaft, whose amount F is equal to the centrifugal force Wa²r/g lb.,
where r is the radius of the mass centre of the weight, and a is its
angular velocity in radians per second. Transferring this force to the
point O, it is equivalent to, (1) a force at O equal and parallel to
F, and, (2) a centrifugal couple of Fa foot-pounds. In order that OX
may be a permanent axis it is necessary that there should be a
sufficient number of weights attached to the shaft and so distributed
that when each is referred to the point O

(1) [Sigma]F = 0 \ (a)
(2) [Sigma]Fa = 0 /

The plane through O to which the shaft is perpendicular is called the
_reference plane_, because all the transferred forces act in that
plane at the point O. The plane through the radius of the weight
containing the axis OX is called the _axial plane_ because it contains
the forces forming the couple due to the transference of F to the
reference plane. Substituting the values of F in (a) the two
conditions become

a²
(1) (W1r1 + W2r2 + W3r3 + ...)--- = 0
g
a² (b)
(2) (W1a1r1 + W2a2r2 + ... )--- = 0
g

In order that these conditions may obtain, the quantities in the
brackets must be zero, since the factor a²/g is not zero. Hence
finally the conditions which must be satisfied by the system of
weights in order that the axis of rotation may be a permanent axis is

(1) (W1r1 + W2r2 + W3r3) = 0
(2) (W1a1r1 + W2a2r2 + W3a3r3) = 0 (c)

It must be remembered that these are all directed quantities, and that
their respective sums are to be taken by drawing vector polygons. In
drawing these polygons the magnitude of the vector of the type Wr is
the product Wr, and the direction of the vector is from the shaft
outwards towards the weight W, parallel to the radius r. For the
vector representing a couple of the type War, if the masses are all on
the same side of the reference plane, the direction of drawing is from
the axis outwards; if the masses are some on one side of the reference
plane and some on the other side, the direction of drawing is from the
axis outwards towards the weight for all masses on the one side, and
from the mass inwards towards the axis for all weights on the other
side, drawing always parallel to the direction defined by the radius
r. The magnitude of the vector is the product War. The conditions (c)
may thus be expressed: first, that the sum of the vectors Wr must form
a closed polygon, and, second, that the sum of the vectors War must
form a closed polygon. The general problem in practice is, given a
system of weights attached to a shaft, to find the respective weights
and positions of two balance weights or counterpoises which must be
added to the system in order to make the shaft a permanent axis, the
planes in which the balance weights are to revolve also being given.
To solve this the reference plane must be chosen so that it coincides
with the plane of revolution of one of the as yet unknown balance
weights. The balance weight in this plane has therefore no couple
corresponding to it. Hence by drawing a couple polygon for the given
weights the vector which is required to close the polygon is at once
found and from it the magnitude and position of the balance weight
which must be added to the system to balance the couples follow at
once. Then, transferring the product Wr corresponding with this
balance weight to the reference plane, proceed to draw the force
polygon. The vector required to close it will determine the second
balance weight, the work may be checked by taking the reference plane
to coincide with the plane of revolution of the second balance weight
and then re-determining them, or by taking a reference plane anywhere
and including the two balance weights trying if condition (c) is
satisfied.