Chapter II: On Applied Dynamics (2)

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).

§ 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_.

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.

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.

§ 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.

§ 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.

§ 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.