§ 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:--
[Illustration: FIG. 126.]
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.
[Illustration: FIG. 127.]
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_.
[Illustration: FIG. 128.]
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.
[Illustration: FIG. 129.]
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.
[Illustration: (From _Balancing of Engines_, by permission of Edward
Arnold.)
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.
When a weight is reciprocated, the equal and opposite force required
for its acceleration at any instant appears as an unbalanced force on
the frame of the machine to which the weight belongs. In the
particular case, where the motion is of the kind known as "simple
harmonic" the disturbing force on the frame due to the reciprocation
of the weight is equal to the component of the centrifugal force in
the line of stroke due to a weight equal to the reciprocated weight
supposed concentrated at the crank pin. Using this principle the
method of finding the balance weights to be added to a given system of
reciprocating weights in order to produce a system of forces on the
frame continuously in equilibrium is exactly the same as that just
explained for a system of revolving weights, because for the purpose
of finding the balance weights each reciprocating weight may be
supposed attached to the crank pin which operates it, thus forming an
equivalent revolving system. The balance weights found as part of the
equivalent revolving system when reciprocated by their respective
crank pins form the balance weights for the given reciprocating
system. These conditions may be exactly realized by a system of
weights reciprocated by slotted bars, the crank shaft driving the
slotted bars rotating uniformly. In practice reciprocation is usually
effected through a connecting rod, as in the case of steam engines. In
balancing the mechanism of a steam engine it is often sufficiently
accurate to consider the motion of the pistons as simple harmonic, and
the effect on the framework of the acceleration of the connecting rod
may be approximately allowed for by distributing the weight of the rod
between the crank pin and the piston inversely as the centre of
gravity of the rod divides the distance between the centre of the
cross head pin and the centre of the crank pin. The moving parts of
the engine are then divided into two complete and independent systems,
namely, one system of revolving weights consisting of crank pins,
crank arms, &c., attached to and revolving with the crank shaft, and a
second system of reciprocating weights consisting of the pistons,
cross-heads, &c., supposed to be moving each in its line of stroke
with simple harmonic motion. The balance weights are to be separately
calculated for each system, the one set being added to the crank shaft
as revolving weights, and the second set being included with the
reciprocating weights and operated by a properly placed crank on the
crank shaft. Balance weights added in this way to a set of
reciprocating weights are sometimes called bob-weights. In the case of
locomotives the balance weights required to balance the pistons are
added as revolving weights to the crank shaft system, and in fact are
generally combined with the weights required to balance the revolving
system so as to form one weight, the counterpoise referred to in the
preceding section, which is seen between the spokes of the wheels of a
locomotive. Although this method balances the pistons in the
horizontal plane, and thus allows the pull of the engine on the train
to be exerted without the variation due to the reciprocation of the
pistons, yet the force balanced horizontally is introduced vertically
and appears as a variation of pressure on the rail. In practice about
two-thirds of the reciprocating weight is balanced in order to keep
this variation of rail pressure within safe limits. The assumption
that the pistons of an engine move with simple harmonic motion is
increasingly erroneous as the ratio of the length of the crank r, to
the length of the connecting rod l increases. A more accurate though
still approximate expression for the force on the frame due to the
acceleration of the piston whose weight is W is given by
W / r \
--- [omega]² r ( cos [theta] + --- cos 2[theta] )
g \ l /
The conditions regulating the balancing of a system of weights
reciprocating under the action of accelerating forces given by the
above expression are investigated in a paper by Otto Schlick, "On
Balancing of Steam Engines," _Trans, Inst. Nav. Arch._ (1900), and in
a paper by W. E. Dalby, "On the Balancing of the Reciprocating Parts
of Engines, including the Effect of the Connecting Rod" (ibid., 1901).
A still more accurate expression than the above is obtained by
expansion in a Fourier series, regarding which and its bearing on
balancing engines see a paper by J. H. Macalpine, "A Solution of the
Vibration Problem" (ibid., 1901). The whole subject is dealt with in a
treatise, _The Balancing of Engines_, by W. E. Dalby (London, 1906).
Most of the original papers on this subject of engine balancing are to
be found in the _Transactions_ of the Institution of Naval Architects.
§ 113.* _Centrifugal Whirling of Shafts._--When a system of revolving
masses is balanced so that the conditions of the preceding section are
fulfilled, the centre of gravity of the system lies on the axis of
revolution. If there is the slightest displacement of the centre of
gravity of the system from the axis of revolution a force acts on the
shaft tending to deflect it, and varies as the deflexion and as the
square of the speed. If the shaft is therefore to revolve stably, this
force must be balanced at any instant by the elastic resistance of the
shaft to deflexion. To take a simple case, suppose a shaft, supported
on two bearings to carry a disk of weight W at its centre, and let the
centre of gravity of the disk be at a distance e from the axis of
rotation, this small distance being due to imperfections of material
or faulty construction. Neglecting the mass of the shaft itself, when
the shaft rotates with an angular velocity a, the centrifugal force
Wa²e/g will act upon the shaft and cause its axis to deflect from the
axis of rotation a distance, y say. The elastic resistance evoked by
this deflexion is proportional to the deflexion, so that if c is a
constant depending upon the form, material and method of support of
the shaft, the following equality must hold if the shaft is to rotate
stably at the stated speed--
W
---(y + e)a² = cy,
g
from which y = Wa²e/(gc - Wa²).
This expression shows that as a increases y increases until when Wa² =
gc, y becomes infinitely large. The corresponding value of a, namely
[root]gc/W, is called the _critical velocity_ of the shaft, and is the
speed at which the shaft ceases to rotate stably and at which
centrifugal whirling begins. The general problem is to find the value
of a corresponding to all kinds of loadings on shafts supported in any
manner. The question was investigated by Rankine in an article in the
_Engineer_ (April 9, 1869). Professor A. G. Greenhill treated the
problem of the centrifugal whirling of an unloaded shaft with
different supporting conditions in a paper "On the Strength of
Shafting exposed both to torsion and to end thrust," _Proc. Inst.
Mech. Eng._ (1883). Professor S. Dunkerley ("On the Whirling and
Vibration of Shafts," _Phil. Trans._, 1894) investigated the question
for the cases of loaded and unloaded shafts, and, owing to the
complication arising from the application of the general theory to the
cases of loaded shafts, devised empirical formulae for the critical
speeds of shafts loaded with heavy pulleys, based generally upon the
following assumption, which is stated for the case of a shaft carrying
one pulley: If N1, N2 be the separate speeds of whirl of the shaft and
pulley on the assumption that the effect of one is neglected when that
of the other is under consideration, then the resulting speed of whirl
due to both causes combined may be taken to be of the form N1N2
[root][(N²1 + N1²)] where N means revolutions per minute. This form is
extended to include the cases of several pulleys on the same shaft.
The interesting and important part of the investigation is that a
number of experiments were made on small shafts arranged in different
ways and loaded in different ways, and the speed at which whirling
actually occurred was compared with the speed calculated from formulae
of the general type indicated above. The agreement between the
observed and calculated values of the critical speeds was in most
cases quite remarkable. In a paper by Dr C. Chree, "The Whirling and
Transverse Vibrations of Rotating Shafts," _Proc. Phys. Soc. Lon._,
vol. 19 (1904); also _Phil. Mag._, vol. 7 (1904), the question is
investigated from a new mathematical point of view, and expressions
for the whirling of loaded shafts are obtained without the necessity
of any assumption of the kind stated above. An elementary presentation
of the problem from a practical point of view will be found in _Steam
Turbines_, by Dr A. Stodola (London, 1905).
[Illustration: FIG. 131.]
§ 114. _Revolving Pendulum. Governors._--In fig. 131 AO represents an
upright axis or spindle; B a weight called a _bob_, suspended by rod
OB from a horizontal axis at O, carried by the vertical axis. When the
spindle is at rest the bob hangs close to it; when the spindle
rotates, the bob, being made to revolve round it, diverges until the
resultant of the centrifugal force and the weight of the bob is a
force acting at O in the direction OB, and then it revolves steadily
in a circle. This combination is called a _revolving_, _centrifugal_,
or _conical pendulum_. Revolving pendulums are usually constructed
with _pairs_ of rods and bobs, as OB, Ob, hung at opposite sides of
the spindle, that the centrifugal forces exerted at the point O may
balance each other.
In finding the position in which the bob will revolve with a given
angular velocity, a, for most practical cases connected with machinery
the mass of the rod may be considered as insensible compared with that
of the bob. Let the bob be a sphere, and from the centre of that
sphere draw BH = y perpendicular to OA. Let OH = z; let W be the
weight of the bob, F its centrifugal force. Then the condition of its
steady revolution is W : F :: z : y; that is to say, y/z = F/W =
ya²/g; consequently
z = g/[alpha]² (69)
Or, if n = [alpha] 2[pi] = [alpha]/6.2832 be the number of turns or
fractions of a turn in a second,
g 0.8165 ft. 9.79771 in. \
z = -------- = ---------- = ----------- > (70)
4[pi]²n² n² n² /
z is called the _altitude of the pendulum_.
[Illustration: FIG. 132.]
If the rod of a revolving pendulum be jointed, as in fig. 132, not to
a point in the vertical axis, but to the end of a projecting arm C,
the position in which the bob will revolve will be the same as if the
rod were jointed to the point O, where its prolongation cuts the
vertical axis.
A revolving pendulum is an essential part of most of the contrivances
called _governors_, for regulating the speed of prime movers, for
further particulars of which see STEAM ENGINE.
_Division 3. Working of Machines of Varying Velocity._
§ 115. _General Principles._--In order that the velocity of every
piece of a machine may be uniform, it is necessary that the forces
acting on each piece should be always exactly balanced. Also, in order
that the forces acting on each piece of a machine may be always
exactly balanced, it is necessary that the velocity of that piece
should be uniform.
An excess of the effort exerted on any piece, above that which is
necessary to balance the resistance, is accompanied with acceleration;
a deficiency of the effort, with retardation.
When a machine is being started from a state of rest, and brought by
degrees up to its proper speed, the effort must be in excess; when it
is being retarded for the purpose of stopping it, the resistance must
be in excess.
An excess of effort above resistance involves an excess of energy
exerted above work performed; that excess of energy is employed in
producing acceleration.
An excess of resistance above effort involves an excess of work
performed above energy expended; that excess of work is performed by
means of the retardation of the machinery.
When a machine undergoes alternate acceleration and retardation, so
that at certain instants of time, occurring at the end of intervals
called _periods_ or _cycles_, it returns to its original speed, then
in each of those periods or cycles the alternate excesses of energy
and of work neutralize each other; and at the end of each cycle the
principle of the equality of energy and work stated in § 87, with all
its consequences, is verified exactly as in the case of machines of
uniform speed.
At intermediate instants, however, other principles have also to be
taken into account, which are deduced from the second law of motion,
as applied to _direct deviation_, or acceleration and retardation.
§ 116. _Energy of Acceleration and Work of Retardation for a Shifting
Body._--Let w be the weight of a body which has a motion of
translation in any path, and in the course of the interval of time
[Delta]t let its velocity be increased at a uniform rate of
acceleration from v1 to v2. The rate of acceleration will be
dv/dt = const. = (v2 - v1)[Delta]t;
and to produce this acceleration a uniform effort will be required,
expressed by
P = w(v2 - v1)g[Delta]t (71)
(The product wv/g of the mass of a body by its velocity is called its
_momentum_; so that the effort required is found by dividing the
increase of momentum by the time in which it is produced.)
To find the _energy_ which has to be exerted to produce the
acceleration from v1 to v2, it is to be observed that the _distance_
through which the effort P acts during the acceleration is
[Delta]s = (v2 + v1)[Delta]t/2;
consequently, the _energy of acceleration_ is
P[Delta]s = w(v2 - v1) (v2 + v1)/2g = w(v2² - v1²)2g, (72)
being proportional to the increase in the square of the velocity, and
_independent of the time_.
In order to produce a _retardation_ from the greater velocity v2 to
the less velocity v1, it is necessary to apply to the body a
_resistance_ connected with the retardation and the time by an
equation identical in every respect with equation (71), except by the
substitution of a resistance for an effort; and in overcoming that
resistance the body _performs work_ to an amount determined by
equation (72), putting Rds for Pas.
§ 117. _Energy Stored and Restored by Deviations of Velocity._--Thus a
body alternately accelerated and retarded, so as to be brought back to
its original speed, performs work during its retardation exactly equal
in amount to the energy exerted upon it during its acceleration; so
that that energy may be considered as _stored_ during the
acceleration, and _restored_ during the retardation, in a manner
analogous to the operation of a reciprocating force (§ 108).
Let there be given the mean velocity V = ½(v2 + v1) of a body whose
weight is w, and let it be required to determine the fluctuation of
velocity v2 - v1, and the extreme velocities v1, v2, which that body
must have, in order alternately to store and restore an amount of
energy E. By equation (72) we have
E = w(v2² - v1²)´2g
which, being divided by V = ½(v2 + v1), gives
E/V = w(v2 - v1)/g;
and consequently
v2 - v1 = gE/Vw (73)
The ratio of this fluctuation to the mean velocity, sometimes called
the unsteadiness of the motion of the body, is
(v2 - v1)V = gE/V²w. (74)
§ 118. _Actual Energy of a Shifting Body._--The energy which must be
exerted on a body of the weight w, to accelerate it from a state of
rest up to a given velocity of translation v, and the equal amount of
work which that body is capable of performing by overcoming resistance
while being retarded from the same velocity of translation v to a
state of rest, is
wv²/2g. (75)
This is called the _actual energy_ of the motion of the body, and is
half the quantity which in some treatises is called vis viva.
The energy stored or restored, as the case may be, by the deviations
of velocity of a body or a system of bodies, is the amount by which
the actual energy is increased or diminished.
§ 119. _Principle of the Conservation of Energy in Machines._--The
following principle, expressing the general law of the action of
machines with a velocity uniform or varying, includes the law of the
equality of energy and work stated in § 89 for machines of uniform
speed.
_In any given interval during the working of a machine, the energy
exerted added to the energy restored is equal to the energy stored
added to the work performed._
§ 120. _Actual Energy of Circular Translation--Moment of
Inertia._--Let a small body of the weight w undergo translation in a
circular path of the radius [rho], with the angular velocity of
deflexion [alpha], so that the common linear velocity of all its
particles is v = [alpha][rho]. Then the actual energy of that body is
wv²/2g = w[alpha]²p²/2g. (76)
By comparing this with the expression for the centrifugal force
(w[alpha]²p/g), it appears that the actual energy of a revolving body
is equal to the potential energy Fp/2 due to the action of the
deflecting force along one-half of the radius of curvature of the path
of the body.
The product wp²/g, by which the half-square of the angular velocity is
multiplied, is called the _moment of inertia_ of the revolving body.
§ 121. _Flywheels._--A flywheel is a rotating piece in a machine,
generally shaped like a wheel (that is to say, consisting of a rim
with spokes), and suited to store and restore energy by the periodical
variations in its angular velocity.
The principles according to which variations of angular velocity store
and restore energy are the same as those of § 117, only substituting
_moment of inertia_ for _mass_, and _angular_ for _linear_ velocity.
Let W be the weight of a flywheel, R its radius of gyration, a2 its
maximum, a1 its minimum, and A = ½([alpha]2 + [alpha]1) its mean
angular velocity. Let
I/S = ([alpha]2 - [alpha]2)/A
denote the _unsteadiness_ of the motion of the flywheel; the
denominator S of this fraction is called the _steadiness_. Let e
denote the quantity by which the energy exerted in each cycle of the
working of the machine alternately exceeds and falls short of the work
performed, and which has consequently to be alternately stored by
acceleration and restored by retardation of the flywheel. The value of
this _periodical excess_ is--
e = R²W ([alpha]2² - [alpha]1²), 2g, (77)
from which, dividing both sides by A², we obtain the following
equations:--
e/A² = R²W/gS \
>. (78)
R²WA²/2g = Se/2 /
The latter of these equations may be thus expressed in words: _The
actual energy due to the rotation of the fly, with its mean angular
velocity, is equal to one-half of the periodical excess of energy
multiplied by the steadiness._
In ordinary machinery S = about 32; in machinery for fine purposes S =
from 50 to 60; and when great steadiness is required S = from 100 to
150.
The periodical excess e may arise either from variations in the effort
exerted by the prime mover, or from variations in the resistance of
the work, or from both these causes combined. When but one flywheel is
used, it should be placed in as direct connexion as possible with that
part of the mechanism where the greatest amount of the periodical
excess originates; but when it originates at two or more points, it is
best to have a flywheel in connexion with each of these points. For
example, in a machine-work, the steam-engine, which is the prime mover
of the various tools, has a flywheel on the crank-shaft to store and
restore the periodical excess of energy arising from the variations in
the effort exerted by the connecting-rod upon the crank; and each of
the slotting machines, punching machines, riveting machines, and other
tools has a flywheel of its own to store and restore energy, so as to
enable the very different resistances opposed to those tools at
different times to be overcome without too great unsteadiness of
motion. For tools performing useful work at intervals, and having only
their own friction to overcome during the intermediate intervals, e
should be assumed equal to the whole work performed at each separate
operation.
§ 122. _Brakes._--A brake is an apparatus for stopping and diminishing
the velocity of a machine by friction, such as the friction-strap
already referred to in § 103. To find the distance s through which a
brake, exerting the friction F, must rub in order to stop a machine
having the total actual energy E at the moment when the brake begins
to act, reduce, by the principles of § 96, the various efforts and
other resistances of the machine which act at the same time with the
friction of the brake to the rubbing surface of the brake, and let R
be their resultant--positive if _resistance_, _negative_ if effort
preponderates. Then
s = E/(F + R). (79)
§ 123. _Energy distributed between two Bodies: Projection and
Propulsion._--Hitherto the effort by which a machine is moved has been
treated as a force exerted between a movable body and a fixed body, so
that the whole energy exerted by it is employed upon the movable body,
and none upon the fixed body. This conception is sensibly realized in
practice when one of the two bodies between which the effort acts is
either so heavy as compared with the other, or has so great a
resistance opposed to its motion, that it may, without sensible error,
be treated as fixed. But there are cases in which the motions of both
bodies are appreciable, and must be taken into account--such as the
projection of projectiles, where the velocity of the _recoil_ or
backward motion of the gun bears an appreciable proportion to the
forward motion of the projectile; and such as the propulsion of
vessels, where the velocity of the water thrown backward by the
paddle, screw or other propeller bears a very considerable proportion
to the velocity of the water moved forwards and sideways by the ship.
In cases of this kind the energy exerted by the effort is
_distributed_ between the two bodies between which the effort is
exerted in shares proportional to the velocities of the two bodies
during the action of the effort; and those velocities are to each
other directly as the portions of the effort unbalanced by resistance
on the respective bodies, and inversely as the weights of the bodies.
To express this symbolically, let W1, W2 be the weights of the bodies;
P the effort exerted between them; S the distance through which it
acts; R1, R2 the resistances opposed to the effort overcome by W1, W2
respectively; E1, E2 the shares of the whole energy E exerted upon W1,
W2 respectively. Then
E : E1 : E2 \
W2(P - R1) + W1(P - R2) P - R1 P - R2 |
:: ----------------------- : ------ : ------ >. (80)
W1W2 W1 W2 /
If R1 = R2, which is the case when the resistance, as well as the
effort, arises from the mutual actions of the two bodies, the above
becomes,
E : E1 : E2 \
:: W1 + W2 : W2 : W1 /, (81)
that is to say, the energy is exerted on the bodies in shares
inversely proportional to their weights; and they receive
accelerations inversely proportional to their weights, according to
the principle of dynamics, already quoted in a note to § 110, that the
mutual actions of a system of bodies do not affect the motion of their
common centre of gravity.
For example, if the weight of a gun be 160 times that of its ball
160/161 of the energy exerted by the powder in exploding will be
employed in propelling the ball, and 1/161 in producing the recoil of
the gun, provided the gun up to the instant of the ball's quitting the
muzzle meets with no resistance to its recoil except the friction of
the ball.
§ 124. _Centre of Percussion._--It is obviously desirable that the
deviations or changes of motion of oscillating pieces in machinery
should, as far as possible, be effected by forces applied at their
centres of percussion.
If the deviation be a _translation_--that is, an equal change of
motion of all the particles of the body--the centre of percussion is
obviously the centre of gravity itself; and, according to the second
law of motion, if dv be the deviation of velocity to be produced in
the interval dt, and W the weight of the body, then
W dv
P = --- · -- (82)
g dt
is the unbalanced effort required.
If the deviation be a rotation about an axis traversing the centre of
gravity, there is no centre of percussion; for such a deviation can
only be produced by a _couple_ of forces, and not by any single force.
Let d[alpha] be the deviation of angular velocity to be produced in
the interval dt, and I the moment of the inertia of the body about an
axis through its centre of gravity; then ½Id([alpha]^2) = I[alpha]
d[alpha] is the variation of the body's actual energy. Let M be the
moment of the unbalanced couple required to produce the deviation;
then by equation 57, § 104, the energy exerted by this couple in the
interval dt is M[alpha] dt, which, being equated to the variation of
energy, gives
d[alpha] R²W d[alpha]
M = I-------- = --- · --------. (83)
dt g dt
R is called the radius of gyration of the body with regard to an axis
through its centre of gravity.
[Illustration: FIG. 133.]
Now (fig. 133) let the required deviation be a rotation of the body BB
about an axis O, not traversing the centre of gravity G, d[alpha]
being, as before, the deviation of angular velocity to be produced in
the interval dt. A rotation with the angular velocity [alpha] about an
axis O may be considered as compounded of a rotation with the same
angular velocity about an axis drawn through G parallel to O and a
translation with the velocity [alpha]. OG, OG being the perpendicular
distance between the two axes. Hence the required deviation may be
regarded as compounded of a deviation of translation dv = OG·d[alpha],
to produce which there would be required, according to equation (82),
a force applied at G perpendicular to the plane OG--
W d[alpha]
P = --- · OG · -------- (84)
g dt
and a deviation d[alpha] of rotation about an axis drawn through G
parallel to O, to produce which there would be required a couple of
the moment M given by equation (83). According to the principles of
statics, the resultant of the force P, applied at G perpendicular to
the plane OG, and the couple M is a force equal and parallel to P, but
applied at a distance GC from G, in the prolongation of the
perpendicular OG, whose value is
GC = M/P = R²/OG. (85)
Thus is determined the position of the centre of percussion C,
corresponding to the axis of rotation O. It is obvious from this
equation that, for an axis of rotation parallel to O traversing C, the
centre of percussion is at the point where the perpendicular OG meets
O.
§ 125.* _To find the moment of inertia of a body about an axis through
its centre of gravity experimentally._--Suspend the body from any
conveniently selected axis O (fig. 48) and hang near it a small plumb
bob. Adjust the length of the plumb-line until it and the body
oscillate together in unison. The length of the plumb-line, measured
from its point of suspension to the centre of the bob, is for all
practical purposes equal to the length OC, C being therefore the
centre of percussion corresponding to the selected axis O. From
equation (85)
R^2 = CG × OG = (OC - OG)OG.
The position of G can be found experimentally; hence OG is known, and
the quantity R² can be calculated, from which and the ascertained
weight W of the body the moment of inertia about an axis through G,
namely, W/g × R², can be computed.
[Illustration: FIG. 134.]
§ 126.* _To find the force competent to produce the instantaneous
acceleration of any link of a mechanism._--In many practical problems
it is necessary to know the magnitude and position of the forces
acting to produce the accelerations of the several links of a
mechanism. For a given link, this force is the resultant of all the
accelerating forces distributed through the substance of the material
of the link required to produce the requisite acceleration of each
particle, and the determination of this force depends upon the
principles of the two preceding sections. The investigation of the
distribution of the forces through the material and the stress
consequently produced belongs to the subject of the STRENGTH OF
MATERIALS (q.v.). Let BK (fig. 134) be any link moving in any manner
in a plane, and let G be its centre of gravity. Then its motion may be
analysed into (1) a translation of its centre of gravity; and (2) a
rotation about an axis through its centre of gravity perpendicular to
its plane of motion. Let [alpha] be the acceleration of the centre of
gravity and let A be the angular acceleration about the axis through
the centre of gravity; then the force required to produce the
translation of the centre of gravity is F = W[alpha]/g, and the couple
required to produce the angular acceleration about the centre of
gravity is M = IA/g, W and I being respectively the weight and the
moment of inertia of the link about the axis through the centre of
gravity. The couple M may be produced by shifting the force F parallel
to itself through a distance x. such that Fx = M. When the link forms
part of a mechanism the respective accelerations of two points in the
link can be determined by means of the velocity and acceleration
diagrams described in § 82, it being understood that the motion of one
link in the mechanism is prescribed, for instance, in the
steam-engine's mechanism that the crank shall revolve uniformly. Let
the acceleration of the two points B and K therefore be supposed
known. The problem is now to find the acceleration [alpha] and A. Take
any pole O (fig. 49), and set out Ob equal to the acceleration of B
and Ok equal to the acceleration of K. Join bk and take the point g so
that KG: GB = kg : gb. Og is then the acceleration of the centre of
gravity and the force F can therefore be immediately calculated. To
find the angular acceleration A, draw kt, bt respectively parallel to
and at right angles to the link KB. Then tb represents the angular
acceleration of the point B relatively to the point K and hence tb/KB
is the value of A, the angular acceleration of the link. Its moment of
inertia about G can be found experimentally by the method explained in
§ 125, and then the value of the couple M can be computed. The value
of x is found immediately from the quotient M/F. Hence the magnitude F
and the position of F relatively to the centre of gravity of the link,
necessary to give rise to the couple M, are known, and this force is
therefore the resultant force required.
[Illustration: FIG. 135.]
§ 127.* _Alternative construction for finding the position of F
relatively to the centre of gravity of the link._--Let B and K be any
two points in the link which for greater generality are taken in fig.
135, so that the centre of gravity G is not in the line joining them.
First find the value of R experimentally. Then produce the given
directions of acceleration of B and K to meet in O; draw a circle
through the three points B, K and O; produce the line joining O and G
to cut the circle in Y; and take a point Z on the line OY so that YG ×
GZ = R². Then Z is a point in the line of action of the force F. This
useful theorem is due to G. T. Bennett, of Emmanuel College,
Cambridge. A proof of it and three corollaries are given in appendix 4
of the second edition of Dalby's _Balancing of Engines_ (London,
1906). It is to be noticed that only the directions of the
accelerations of two points are required to find the point Z.
For an example of the application of the principles of the two
preceding sections to a practical problem see _Valve and Valve Gear
Mechanisms_, by W. E. Dalby (London, 1906), where the inertia stresses
brought upon the several links of a Joy valve gear, belonging to an
express passenger engine of the Lancashire & Yorkshire railway, are
investigated for an engine-speed of 68 m. an hour.
[Illustration: FIG. 136.]
§ 128.* _The Connecting Rod Problem._--A particular problem of
practical importance is the determination of the force producing the
motion of the connecting rod of a steam-engine mechanism of the usual
type. The methods of the two preceding sections may be used when the
acceleration of two points in the rod are known. In this problem it is
usually assumed that the crank pin K (fig. 136) moves with uniform
velocity, so that if [alpha] is its angular velocity and r its radius,
the acceleration is [alpha]²r in a direction along the crank arm from
the crank pin to the centre of the shaft. Thus the acceleration of one
point K is known completely. The acceleration of a second point,
usually taken at the centre of the crosshead pin, can be found by the
principles of § 82, but several special geometrical constructions have
been devised for this purpose, notably the construction of Klein,[4]
discovered also independently by Kirsch.[5] But probably the most
convenient is the construction due to G. T. Bennett[6] which is as
follows: Let OK be the crank and KB the connecting rod. On the
connecting rod take a point L such that KL × KB = KO². Then, the crank
standing at any angle with the line of stroke, draw LP at right angles
to the connecting rod, PN at right angles to the line of stroke OB and
NA at right angles to the connecting rod; then AO is the acceleration
of the point B to the scale on which KO represents the acceleration of
the point K. The proof of this construction is given in _The Balancing
of Engines_.
The finding of F may be continued thus: join AK, then AK is the
acceleration image of the rod, OKA being the acceleration diagram.
Through G, the centre of gravity of the rod, draw Gg parallel to the
line of stroke, thus dividing the image at g in the proportion that
the connecting rod is divided by G. Hence Og represents the
acceleration of the centre of gravity and, the weight of the
connecting rod being ascertained, F can be immediately calculated. To
find a point in its line of action, take a point Q on the rod such
that KG × GQ = R², R having been determined experimentally by the
method of § 125; join G with O and through Q draw a line parallel to
BO to cut GO in Z. Z is a point in the line of action of the resultant
force F; hence through Z draw a line parallel to Og. The force F acts
in this line, and thus the problem is completely solved. The above
construction for Z is a corollary of the general theorem given in §
127.
§ 129. _Impact._ Impact or collision is a pressure of short duration
exerted between two bodies.
The effects of impact are sometimes an alteration of the distribution
of actual energy between the two bodies, and always a loss of a
portion of that energy, depending on the imperfection of the
elasticity of the bodies, in permanently altering their figures, and
producing heat. The determination of the distribution of the actual
energy after collision and of the loss of energy is effected by means
of the following principles:--
I. The motion of the common centre of gravity of the two bodies is
unchanged by the collision.
II. The loss of energy consists of a certain proportion of that part
of the actual energy of the bodies which is due to their motion
relatively to their common centre of gravity.
Unless there is some special reason for using impact in machines, it
ought to be avoided, on account not only of the waste of energy which
it causes, but from the damage which it occasions to the frame and
mechanism. (W. J. M. R.; W. E. D.)
FOOTNOTES:
[1] In view of the great authority of the author, the late Professor
Macquorn Rankine, it has been thought desirable to retain the greater
part of this article as it appeared in the 9th edition of the
_Encyclopaedia Britannica_. Considerable additions, however, have
been introduced in order to indicate subsequent developments of the
subject; the new sections are numbered continuously with the old, but
are distinguished by an asterisk. Also, two short chapters which
concluded the original article have been omitted--ch. iii., "On
Purposes and Effects of Machines," which was really a classification
of machines, because the classification of Franz Reuleaux is now
usually followed, and ch. iv., "Applied Energetics, or Theory of
Prime Movers," because its subject matter is now treated in various
special articles, e.g. Hydraulics, Steam Engine, Gas Engine, Oil
Engine, and fully developed in Rankine's The Steam Engine and Other
Prime Movers (London, 1902). (Ed. _E.B._)
[2] Since the relation discussed in § 7 was enunciated by Rankine, an
enormous development has taken place in the subject of Graphic
Statics, the first comprehensive textbook on the subject being _Die
Graphische Statik_ by K. Culmann, published at Zürich in 1866. Many
of the graphical methods therein given have now passed into the
textbooks usually studied by engineers. One of the most beautiful
graphical constructions regularly used by engineers and known as "the
method of reciprocal figures" is that for finding the loads supported
by the several members of a braced structure, having given a system
of external loads. The method was discovered by Clerk Maxwell, and
the complete theory is discussed and exemplified in a paper "On
Reciprocal Figures, Frames and Diagrams of Forces," _Trans. Roy. Soc.
Ed._, vol. xxvi. (1870). Professor M. W. Crofton read a paper on
"Stress-Diagrams in Warren and Lattice Girders" at the meeting of the
Mathematical Society (April 13, 1871), and Professor O. Henrici
illustrated the subject by a simple and ingenious notation. The
application of the method of reciprocal figures was facilitated by a
system of notation published in _Economics of Construction in
relation to framed Structures_, by Robert H. Bow (London, 1873). A
notable work on the general subject is that of Luigi Cremona,
translated from the Italian by Professor T. H. Beare (Oxford, 1890),
and a discussion of the subject of reciprocal figures from the
special point of view of the engineering student is given in _Vectors
and Rotors_ by Henrici and Turner (London, 1903). See also above
under "_Theoretical Mechanics_," Part 1. § 5.
[3] This is a particular case of a more general principle, that _the
motion of the centre of gravity of a body is not affected by the
mutual actions of its parts_.
[4] J. F. Klein, "New Constructions of the Force of Inertia of
Connecting Rods and Couplers and Constructions of the Pressures on
their Pins," _Journ. Franklin Inst._, vol. 132 (Sept. and Oct.,
1891).
[5] Prof. Kirsch, "Über die graphische Bestimmung der
Kolbenbeschleunigung," _Zeitsch. Verein deutsche Ingen_. (1890), p.
1320.
[6] Dalby, _The Balancing of Engines_ (London, 1906), app. 1.
MECHANICVILLE, a village of Saratoga county, New York, U.S.A., on the
west bank of the Hudson River, about 20 m. N. of Albany; on the Delaware
& Hudson and Boston & Maine railways. Pop. (1900), 4695 (702
foreign-born); (1905, state census), 5877; (1910) 6,634. It lies partly
within Stillwater and partly within Half-Moon townships, in the
bottom-lands at the mouth of the Anthony Kill, about 1-1/2 m. S. of the
mouth of the Hoosick River. On the north and south are hills reaching a
maximum height of 200 ft. There is ample water power, and there are
manufactures of paper, sash and blinds, fibre, &c. From a dam here power
is derived for the General Electric Company at Schenectady. The first
settlement in this vicinity was made in what is now Half-Moon township
about 1680. Mechanicville (originally called Burrow) was chartered by
the county court in 1859, and incorporated as a village in 1870. It was
the birthplace of Colonel Ephraim Elmer Ellsworth (1837-1861), the first
Federal officer to lose his life in the Civil War.
MECHITHARISTS, a congregation of Armenian monks in communion with the
Church of Rome. The founder, Mechithar, was born at Sebaste in Armenia,
1676. He entered a monastery, but under the influence of Western
missionaries he became possessed with the idea of propagating Western
ideas and culture in Armenia, and of converting the Armenian Church from
its monophysitism and uniting it to the Latin Church. Mechithar set out
for Rome in 1695 to make his ecclesiastical studies there, but he was
compelled by illness to abandon the journey and return to Armenia. In
1696 he was ordained priest and for four years worked among his people.
In 1700 he went to Constantinople and began to gather disciples around
him. Mechithar formally joined the Latin Church, and in 1701, with
sixteen companions, he formed a definitely religious institute of which
he became the superior. Their Uniat propaganda encountered the
opposition of the Armenians and they were compelled to move to the
Morea, at that time Venetian territory, and there built a monastery,
1706. On the outbreak of hostilities between the Turks and Venetians
they migrated to Venice, and the island of St Lazzaro was bestowed on
them, 1717. This has since been the headquarters of the congregation,
and here Mechithar died in 1749, leaving his institute firmly
established. The rule followed at first was that attributed to St
Anthony; but when they settled in the West modifications from the
Benedictine rule were introduced, and the Mechitharists are numbered
among the lesser orders affiliated to the Benedictines. They have ever
been faithful to their founder's programme. Their work has been
fourfold: (1) they have brought out editions of important patristic
works, some Armenian, others translated into Armenian from Greek and
Syriac originals no longer extant; (2) they print and circulate Armenian
literature among the Armenians, and thereby exercise a powerful
educational influence; (3) they carry on schools both in Europe and
Asia, in which Uniat Armenian boys receive a good secondary education;
(4) they work as Uniat missioners in Armenia. The congregation is
divided into two branches, the head houses being at St Lazzaro and
Vienna. They have fifteen establishments in various places in Asia Minor
and Europe. There are some 150 monks, all Armenians; they use the
Armenian language and rite in the liturgy.
See _Vita del servo di Dio Mechitar_ (Venice, 1901); E. Boré,
_Saint-Lazare_ (1835); Max Heimbucher, _Orden u. Kongregationen_
(1907) I. § 37; and the articles in Wetzer u. Welte, _Kirchenlexicon_
(ed. 2) and Herzog, _Realencyklopädie_ (ed. 3), also articles by
Sargisean, a Mechitharist, in _Rivista storica benedettina_ (1906),
"La Congregazione Mechitarista." (E. C. B.)
MECKLENBURG, a territory in northern Germany, on the Baltic Sea,
extending from 53° 4´ to 54° 22´ N. and from 10° 35´ to 13° 57´ E.,
unequally divided into the two grand duchies of Mecklenburg-Schwerin and
Mecklenburg-Strelitz.
MECKLENBURG-SCHWERIN is bounded N. by the Baltic Sea, W. by the
principality of Ratzeburg and Schleswig-Holstein, S. by Brandenburg and
Hanover, and E. by Pomerania and Mecklenburg-Strelitz. It embraces the
duchies of Schwerin and Güstrow, the district of Rostock, the
principality of Schwerin, and the barony of Wismar, besides several
small enclaves (Ahrensberg, Rosson, Tretzeband, &c.) in the adjacent
territories. Its area is 5080 sq. m. Pop. (1905), 625,045.
MECKLENBURG-STRELITZ consists of two detached parts, the duchy of
Strelitz on the E. of Mecklenburg-Schwerin, and the principality of
Ratzeburg on the W. The first is bounded by Mecklenburg-Schwerin,
Pomerania and Brandenburg, the second by Mecklenburg-Schwerin,
Lauenburg, and the territory of the free town of Lübeck. Their joint
area is 1130 sq. m. Pop. (1905), 103,451.
Mecklenburg lies wholly within the great North-European plain, and its
flat surface is interrupted only by one range of low hills,
intersecting the country from south-east to north-west, and forming
the watershed between the Baltic Sea and the Elbe. Its highest point,
the Helpter Berg, is 587 ft. above sea-level. The coast-line runs for
65 m. along the Baltic (without including indentations), for the most
part in flat sandy stretches covered with dunes. The chief inlets are
Wismar Bay, the Salzhaff, and the roads of Warnemünde. The rivers are
numerous though small; most of them are affluents of the Elbe, which
traverses a small portion of Mecklenburg. Several are navigable, and
the facilities for inland water traffic are increased by canals. Lakes
are numerous; about four hundred, covering an area of 500 sq. m., are
reckoned in the two duchies. The largest is Lake Müritz, 52 sq. m. in
extent. The climate resembles that of Great Britain, but the winters
are generally more severe; the mean annual temperature is 48° F., and
the annual rainfall is about 28 in. Although there are long stretches
of marshy moorland along the coast, the soil is on the whole
productive. About 57% of the total area of Mecklenburg-Schwerin
consists of cultivated land, 18% of forest, and 13% of heath and
pasture. In Mecklenburg-Strelitz the corresponding figures are 47, 21
and 10%. Agriculture is by far the most important industry in both
duchies. The chief crops are rye, oats, wheat, potatoes and hay.
Smaller areas are devoted to maize, buckwheat, pease, rape, hemp,
flax, hops and tobacco. The extensive pastures support large herds of
sheep and cattle, including a noteworthy breed of merino sheep. The
horses of Mecklenburg are of a fine sturdy quality and highly
esteemed. Red deer, wild swine and various other game are found in the
forests. The industrial establishments include a few iron-foundries,
wool-spinning mills, carriage and machine factories, dyeworks,
tanneries, brick-fields, soap-works, breweries, distilleries, numerous
limekilns and tar-boiling works, tobacco and cigar factories, and
numerous mills of various kinds. Mining is insignificant, though a
fair variety of minerals is represented in the district. Amber is
found on and near the Baltic coast. Rostock, Warnemünde and Wismar are
the principal commercial centres. The chief exports are grain and
other agricultural produce, live stock, spirits, wood and wool; the
chief imports are colonial produce, iron, coal, salt, wine, beer and
tobacco. The horse and wool markets of Mecklenburg are largely
attended by buyers from various parts of Germany. Fishing is carried
on extensively in the numerous inland lakes.
In 1907 the grand dukes of both duchies promised a constitution to
their subjects. The duchies had always been under a government of
feudal character, the grand dukes having the executive entirely in
their hands (though acting through ministers), while the duchies
shared a diet (_Landtag_), meeting for a short session each year, and
at other times represented by a committee, and consisting of the
proprietors of knights' estates (_Rittergüter_), known as the
_Ritterschaft_, and the _Landschaft_ or burgomasters of certain towns.
Mecklenburg-Schwerin returns six members to the Reichstag and
Mecklenburg-Strelitz one member.
In Mecklenburg-Schwerin the chief towns are Rostock (with a
university), Schwerin, and Wismar the capital. The capital of
Mecklenburg-Strelitz is Neu-Strelitz. The peasantry of Mecklenburg
retain traces of their Slavonic origin, especially in speech, but
their peculiarities have been much modified by amalgamation with
German colonists. The townspeople and nobility are almost wholly of
Saxon strain. The slowness of the increase in population is chiefly
accounted for by emigration.
_History._--The Teutonic peoples, who in the time of Tacitus occupied
the region now known as Mecklenburg, were succeeded in the 6th century
by some Slavonic tribes, one of these being the Obotrites, whose chief
fortress was Michilenburg, the modern Mecklenburg, near Wismar; hence
the name of the country. Though partly subdued by Charlemagne towards
the close of the 8th century, they soon regained their independence, and
until the 10th century no serious effort was made by their Christian
neighbours to subject them. Then the German king, Henry the Fowler,
reduced the Slavs of Mecklenburg to obedience and introduced
Christianity among them. During the period of weakness through which the
German kingdom passed under the later Ottos, however, they wrenched
themselves free from this bondage; the 11th and the early part of the
12th century saw the ebb and flow of the tide of conquest, and then came
the effective subjugation of Mecklenburg by Henry the Lion, duke of
Saxony. The Obotrite prince Niklot was killed in battle in 1160 whilst
resisting the Saxons, but his son Pribislaus (d. 1178) submitted to
Henry the Lion, married his daughter to the son of the duke, embraced
Christianity, and was permitted to retain his office. His descendants
and successors, the present grand dukes of Mecklenburg, are the only
ruling princes of Slavonic origin in Germany. Henry the Lion introduced
German settlers and restored the bishoprics of Ratzeburg and Schwerin;
in 1170 the emperor Frederick I. made Pribislaus a prince of the empire.
From 1214 to 1227 Mecklenburg was under the supremacy of Denmark; then,
in 1229, after it had been regained by the Germans, there took place the
first of the many divisions of territory which with subsequent reunions
constitute much of its complicated history. At this time the country was
divided between four princes, grandsons of duke Henry Borwin, who had
died two years previously. But in less than a century the families of
two of these princes became extinct, and after dividing into three
branches a third family suffered the same fate in 1436. There then
remained only the line ruling in Mecklenburg proper, and the princes of
this family, in addition to inheriting the lands of their dead kinsmen,
made many additions to their territory, including the counties of
Schwerin and of Strelitz. In 1352 the two princes of this family made a
division of their lands, Stargard being separated from the rest of the
country to form a principality for John (d. 1393), but on the extinction
of his line in 1471 the whole of Mecklenburg was again united under a
single ruler. One member of this family, Albert (c. 1338-1412), was king
of Sweden from 1364 to 1389. In 1348 the emperor Charles IV. had raised
Mecklenburg to the rank of a duchy, and in 1418 the university of
Rostock was founded.
The troubles which arose from the rivalry and jealousy of two or more
joint rulers incited the prelates, the nobles and the burghers to form a
union among themselves, and the results of this are still visible in the
existence of the _Landesunion_ for the whole country which was
established in 1523. About the same time the teaching of Luther and the
reformers was welcomed in Mecklenburg, although Duke Albert (d. 1547)
soon reverted to the Catholic faith; in 1549 Lutheranism was recognized
as the state religion; a little later the churches and schools were
reformed and most of the monasteries were suppressed. A division of the
land which took place in 1555 was of short duration, but a more
important one was effected in 1611, although Duke John Albert I. (d.
1576) had introduced the principle of primogeniture and had forbidden
all further divisions of territory. By this partition John Albert's
grandson Adolphus Frederick I. (d. 1658) received Schwerin, and another
grandson John Albert II. (d. 1636) received Güstrow. The town of
Rostock "with its university and high court of justice" was declared to
be common property, while the Diet or _Landtag_ also retained its joint
character, its meetings being held alternately at Sternberg and at
Malchin.
During the early part of the Thirty Years' War the dukes of
Mecklenburg-Schwerin and Mecklenburg-Güstrow were on the Protestant
side, but about 1627 they submitted to the emperor Ferdinand II. This
did not prevent Ferdinand from promising their land to Wallenstein, who,
having driven out the dukes, was invested with the duchies in 1629 and
ruled them until 1631. In this year the former rulers were restored by
Gustavus Adolphus of Sweden, and in 1635 they came to terms with the
emperor and signed the peace of Prague, but their land continued to be
ravaged by both sides until the conclusion of the war. In 1648 by the
Treaty of Westphalia, Wismar and some other parts of Mecklenburg were
surrendered to Sweden, the recompense assigned to the duchies including
the secularized bishoprics of Schwerin and of Ratzeburg. The sufferings
of the peasants in Mecklenburg during the Thirty Years' War were not
exceeded by those of their class in any other part of Germany; most of
them were reduced to a state of serfdom and in some cases whole villages
vanished. Christian Louis who ruled Mecklenburg-Schwerin from 1658 until
his death in 1692 was, like his father Adolphus Frederick, frequently at
variance with the estates of the land and with members of his family. He
was a Roman Catholic and a supporter of Louis XIV., and his country
suffered severely during the wars waged by France and her allies in
Germany.
In June 1692 when Christian Louis died in exile and without sons, a
dispute arose about the succession to his duchy between his brother
Adolphus Frederick and his nephew Frederick William. The emperor and the
rulers of Sweden and of Brandenburg took part in this struggle which was
intensified when, three years later, on the death of Duke Gustavus
Adolphus, the family ruling over Mecklenburg-Güstrow became extinct. At
length the partition Treaty of Hamburg was signed on the 8th of March
1701, and a new division of the country was made. Mecklenburg was
divided between the two claimants, the shares given to each being
represented by the existing duchies of Mecklenburg-Schwerin, the part
which fell to Frederick William, and Mecklenburg-Strelitz, the share of
Adolphus Frederick. At the same time the principle of primogeniture was
again asserted, and the right of summoning the joint _Landtag_ was
reserved to the ruler of Mecklenburg-Schwerin.
Mecklenburg-Schwerin began its existence by a series of constitutional
struggles between the duke and the nobles. The heavy debt incurred by
Duke Charles Leopold (d. 1747), who had joined Russia in a war against
Sweden, brought matters to a crisis; the emperor Charles VI. interfered
and in 1728 the imperial court of justice declared the duke incapable of
governing and his brother Christian Louis was appointed administrator of
the duchy. Under this prince, who became ruler _de jure_ in 1747, there
was signed in April 1755 the convention of Rostock by which a new
constitution was framed for the duchy. By this instrument all power was
in the hands of the duke, the nobles and the upper classes generally,
the lower classes being entirely unrepresented. During the Seven Years'
War Duke Frederick (d. 1785) took up a hostile attitude towards
Frederick the Great, and in consequence Mecklenburg was occupied by
Prussian troops, but in other ways his rule was beneficial to the
country. In the early years of the French revolutionary wars Duke
Frederick Francis I. (1756-1837) remained neutral, and in 1803 he
regained Wismar from Sweden, but in 1806 his land was overrun by the
French and in 1808 he joined the Confederation of the Rhine. He was the
first member of the confederation to abandon Napoleon, to whose armies
he had sent a contingent, and in 1813-1814 he fought against France. In
1815 he joined the Germanic Confederation (Bund) and took the title of
grand duke. In 1819 serfdom was abolished in his dominions. During the
movement of 1848 the duchy witnessed a considerable agitation in favour
of a more liberal constitution, but in the subsequent reaction all the
concessions which had been made to the democracy were withdrawn and
further restrictive measures were introduced in 1851 and 1852.
Mecklenburg-Strelitz adopted the constitution of the sister duchy by an
act of September 1755. In 1806 it was spared the infliction of a French
occupation through the good offices of the king of Bavaria; in 1808 its
duke, Charles (d. 1816), joined the confederation of the Rhine, but in
1813 he withdrew therefrom. Having been a member of the alliance against
Napoleon he joined the Germanic confederation in 1815 and assumed the
title of grand duke.
In 1866 both the grand dukes of Mecklenburg joined the North German
confederation and the _Zollverein_, and began to pass more and more
under the influence of Prussia, who in the war with Austria had been
aided by the soldiers of Mecklenburg-Schwerin. In the Franco-German War
also Prussia received valuable assistance from Mecklenburg, Duke
Frederick Francis II. (1823-1883), an ardent advocate of German unity,
holding a high command in her armies. In 1871 the two grand duchies
became states of the German Empire. There was now a renewal of the
agitation for a more democratic constitution, and the German Reichstag
gave some countenance to this movement. In 1897 Frederick Francis IV.
(b. 1882) succeeded his father Frederick Francis III. (1851-1897) as
grand duke of Mecklenburg-Schwerin, and in 1904 Adolphus Frederick (b.
1848) a son of the grand duke Frederick William (1819-1904) and his wife
Augusta Carolina, daughter of Adolphus Frederick, duke of Cambridge,
became grand duke of Mecklenburg-Strelitz. The grand dukes still style
themselves princes of the Wends.
See F. A. Rudloff, _Pragmatisches Handbuch der mecklenburgischen
Geschichte_ (Schwerin, 1780-1822); C. C. F. von Lützow, _Versuch einer
pragmatischen Geschichte von Mecklenburg_ (Berlin, 1827-1835);
_Mecklenburgische Geschichte in Einzeldarstellungen_, edited by R.
Beltz, C. Beyer, W. P. Graff and others; C. Hegel, _Geschichte der
mecklenburgischen Landstände bis 1555_ (Rostock, 1856); A. Mayer,
_Geschichte des Grossherzogtums Mecklenburg-Strelitz 1816-1890_ (New
Strelitz, 1890); Tolzien, _Die Grossherzöge von Mecklenburg-Schwerin_
(Wismar, 1904); Lehsten, _Der Adel Mecklenburgs seit dem
landesgrundgesetslichen Erbvergleich_ (Rostock, 1864); the
_Mecklenburgisches Urkundenbuch_ in 21 vols. (Schwerin, 1873-1903);
the _Jahrbücher des Vereins für mecklenburgische Geschichte und
Altertumskunde_ (Schwerin, 1836 fol.); and W. Raabe, _Mecklenburgische
Vaterlandskunde_ (Wismar, 1894-1896); von Hirschfeld, _Friedrich Franz
II., Grossherzog von Mecklenburg-Schwerin und seine Vorgänger_
(Leipzig, 1891); Volz, _Friedrich Franz II._ (Wismar, 1893); C.
Schröder, _Friedrich Franz III._ (Schwerin, 1898); Bartold, _Friedrich
Wilhelm, Grossherzog von Mecklenburg-Strelitz und Augusta Carolina_
(New Strelitz, 1893); and H. Sachsse, _Mecklenburgische Urkunden und
Daten_ (Rostock, 1900).