Encyclopaedia Britannica, 11th Edition, "Dyer, Sir Edward" to "Echidna"
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Title: Encyclopaedia Britannica, 11th Edition, "Dyer, Sir Edward" to "Echidna"
Author: Various
Release date: January 8, 2011 [eBook #34878]
Language: English
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Distributed Proofreading Team at http://www.pgdp.net
Transcriber's notes:
(1) Numbers following letters (without space) like C2 were originally
printed in subscript. Letter subscripts are preceded by an
underscore, like C_n.
(2) Characters following a carat (^) were printed in superscript.
(3) Side-notes were relocated to function as titles of their respective
paragraphs.
(4) Macrons and breves above letters and dots below letters were not
inserted.
(5) dP stands for the partial-derivative symbol, or curled 'd'.
(6) [oo] stands for the infinity symbol, and [int] for the integral
symbol.
(7) Letters followed with a grave accent "`" have originally dots above.
(8) The following typographical errors have been corrected:
ARTICLE DYER, JOHN: "His poems were collected by Dodsley in 1770,
and by Mr Edward Thomas in 1903 for the Welsh Library, vol. iv."
'poems' amended from 'peoms'.
ARTICLE EAR: "The membranous semicircular canals are very much
smaller in section than the bony; in the ampulla of each is a
ridge..." 'the' amended from 'tbe'.
ARTICLE EARTH, FIGURE OF THE: "O. Callandreau, 'Memoire sur la
theorie de la figure des planetes,' Ann. obs. de Paris (1889);..."
'Callandreau' amended from 'Callendreau'.
ARTICLE EATON, THEOPHILUS: "In October 1639 a form of government
was adopted, based on the Mosaic Law, and Eaton was elected
governor..." 'Mosaic' amended from 'Mosiac'.
ARTICLE ECCLESIASTES: "A particular instance is mentioned (ix.
13-15) of a beleaguered city saved by a wise man; but the man
happened to be poor, and no one remembered him." 'beleaguered'
amended from 'beleagured'.
ARTICLE ECCLESIASTES: "Such assertions as those of ii. 26 (God
gives joy to him who pleases him, and makes the sinner toil to lay
up for the latter),..." 'and' amended from 'amd'.
ARTICLE ECCLESIASTES: "This disagreement comes largely from the
attempts made to find definitely expressed Greek philosophical
dogmas in the book; such formulas it has not, but the general air
of Greek reflection seems unmistakable. The scepticism of Koheleth
differs from that of Job in quality and scope..." 'the' originally
repeated twice.
ARTICLE ECCLESIASTICAL JURISDICTION: "In the first case, they may
be punished by the ordinary of the place, acting as delegate of the
pope without special appointment (Conc. Trid. Sess. vi. c. 3)."
'special' amended from 'speical'.
ENCYCLOPAEDIA BRITANNICA
A DICTIONARY OF ARTS, SCIENCES, LITERATURE
AND GENERAL INFORMATION
ELEVENTH EDITION
VOLUME VIII, SLICE IX
Dyer to Echidna
ARTICLES IN THIS SLICE:
DYER, SIR EDWARD EAST LIVERPOOL
DYER, JOHN EAST LONDON
DYER, THOMAS HENRY EASTON
DYMOKE EAST ORANGE
DYNAMICS EASTPORT
DYNAMITE EAST PROVIDENCE
DYNAMO EAST PRUSSIA
DYNAMOMETER EASTWICK, EDWARD BACKHOUSE
DYNASTY EATON, DORMAN BRIDGMAN
DYSART EATON, MARGARET O'NEILL
DYSENTERY EATON, THEOPHILUS
DYSPEPSIA EATON, WILLIAM
DYSTELEOLOGY EATON, WYATT
DZUNGARIA EAU CLAIRE
E EAU DE COLOGNE
EA EAUX-BONNES
EABANI EAVES
EACHARD, JOHN EAVESDRIP
EADBALD EBBW VALE
EADIE, JOHN EBEL, HERMANN WILHELM
EADMER EBEL, JOHANN GOTTFRIED
EADS, JAMES BUCHANAN EBER, PAUL
EAGLE EBERBACH (town of Germany)
EAGLEHAWK EBERBACH (monastery of Germany)
EAGRE EBERHARD
EAKINS, THOMAS EBERHARD, CHRISTIAN AUGUST GOTTLOB
EALING EBERHARD, JOHANN AUGUSTUS
EAR EBERLIN, JOHANN ERNST
EARL EBERS, GEORG MORITZ
EARLE, JOHN EBERSWALDE
EARLE, RALPH EBERT, FRIEDRICH ADOLF
EARL MARSHAL EBINGEN
EARLOM, RICHARD EBIONITES
EARLSTON EBNER-ESCHENBACH, MARIE
EARLY, JUBAL ANDERSON EBOLI
EARLY ENGLISH PERIOD EBONY
EARN EBRARD, JOHANNES HEINRICH AUGUST
EARNEST EBRO
EAR-RING EBROIN
EARTH EBURACUM
EARTH, FIGURE OF THE ECA DE QUEIROZ, JOSE MARIA
EARTH CURRENTS ECARTE
EARTH-NUT ECBATANA
EARTH PILLAR ECCARD, JOHANN
EARTHQUAKE ECCELINO DA ROMANO
EARTH-STAR ECCENTRIC
EARTHWORM ECCHELLENSIS, ABRAHAM
EARWIG ECCLES
EASEMENT ECCLESFIELD
EAST, ALFRED ECCLESHALL
EAST ANGLIA ECCLESIA
EASTBOURNE ECCLESIASTES
EAST CHICAGO ECCLESIASTICAL COMMISSIONERS
EASTER ECCLESIASTICAL JURISDICTION
EASTER ISLAND ECCLESIASTICAL LAW
EASTERN BENGAL AND ASSAM ECCLESIASTICUS
EASTERN QUESTION, THE ECGBERT (king of the West Saxons)
EAST GRINSTEAD ECGBERT (archbishop of York)
EAST HAM ECGFRITH
EASTHAMPTON ECGONINE
EAST HAMPTON ECHEGARAY Y EIZAGUIRRE, JOSE
EAST INDIA COMPANY ECHELON
EAST INDIES ECHIDNA
EASTLAKE, SIR CHARLES LOCK
DYER, SIR EDWARD (d. 1607), English courtier and poet, son of Sir Thomas
Dyer, Kt., was born at Sharpham Park, Somersetshire. He was educated,
according to Anthony a Wood, either at Balliol College or at Broadgates
Hall, Oxford. He left the university without taking a degree, and after
some time spent abroad appeared at Queen Elizabeth's court. His first
patron was the earl of Leicester, who seems to have thought of putting
him forward as a rival to Sir Christopher Hatton in the queen's favour.
He is mentioned by Gabriel Harvey with Sidney as one of the ornaments of
the court. Sidney in his will desired that his books should be divided
between Fulke Greville (Lord Brooke) and Dyer. He was employed by
Elizabeth on a mission (1584) to the Low Countries, and in 1589 was sent
to Denmark. In a commission to inquire into manors unjustly alienated
from the crown in the west country he did not altogether please the
queen, but he received a grant of some forfeited lands in Somerset in
1588. He was knighted and made chancellor of the order of the Garter in
1596. William Oldys says of him that he "would not stoop to fawn," and
some of his verses seem to show that the exigencies of life at court
oppressed him. He was buried at St Saviour's, Southwark, on the 11th of
May 1607. Wood says that many esteemed him to be a Rosicrucian, and that
he was a firm believer in alchemy. He had a great reputation as a poet
among his contemporaries, but very little of his work has survived.
Puttenham in the _Arte of English Poesie_ speaks of "Maister Edward
Dyar, for Elegie most sweete, solempne, and of high conceit." One of the
poems universally accepted as his is "My Mynde to me a kingdome is."
Among the poems in _England's Helicon_ (1600), signed S.E.D., and
included in Dr A.B. Grosart's collection of Dyer's works (_Miscellanies
of the Fuller Worthies Library_, vol. iv., 1876) is the charming
pastoral "My Phillis hath the morninge sunne," but this comes from the
_Phillis_ of Thomas Lodge. Grosart also prints a prose tract entitled
_The Prayse of Nothing_ (1585). The _Sixe Idillia_ from Theocritus,
reckoned by J.P. Collier among Dyer's works, were dedicated to, not
written by, him.
DYER, JOHN (c. 1700-1758), British poet, the son of a solicitor, was
born in 1699 or 1700 at Aberglasney, in Carmarthenshire. He was sent to
Westminster school and was destined for the law, but on his father's
death he began to study painting. He wandered about South Wales,
sketching and occasionally painting portraits. In 1726 his first poem,
_Grongar Hill_, appeared in a miscellany published by Richard Savage,
the poet. It was an irregular ode in the so-called Pindaric style, but
Dyer entirely rewrote it into a loose measure of four cadences, and
printed it separately in 1727. It had an immediate and brilliant
success. _Grongar Hill_, as it now stands, is a short poem of only 150
lines, describing in language of much freshness and picturesque charm
the view from a hill overlooking the poet's native vale of Towy. A visit
to Italy bore fruit in _The Ruins of Rome_ (1740), a descriptive piece
in about 600 lines of Miltonic blank verse. He was ordained priest in
1741, and held successively the livings of Calthorp in Leicestershire,
Belchford (1751), Coningsby (1752), and Kirby-on-Bane (1756), the last
three being Lincolnshire parishes. He married, in 1741, a Miss Ensor,
said to be descended from the brother of Shakespeare. In 1757 he
published his longest work, the didactic blank-verse epic of _The
Fleece_, in four books, discoursing of the tending of sheep, of the
shearing and preparation of the wool, of weaving, and of trade in
woollen manufactures. The town took no interest in it, and Dodsley
facetiously prophesied that "Mr Dyer would be buried in woollen." He
died at Coningsby of consumption, on the 15th of December 1758.
His poems were collected by Dodsley in 1770, and by Mr Edward Thomas
in 1903 for the _Welsh Library_, vol. iv.
DYER, THOMAS HENRY (1804-1888), English historical and antiquarian
writer, was born in London on the 4th of May 1804. He was originally
intended for a business career, and for some time acted as clerk in a
West India house; but finding his services no longer required after the
passing of the Negro Emancipation Act, he decided to devote himself to
literature. In 1850 he published the _Life of Calvin_, a conscientious
and on the whole impartial work, though the character of Calvin is
somewhat harshly drawn, and his influence in the religious world
generally is insufficiently appreciated. Dyer's first historical work
was the _History of Modern Europe_ (1861-1864; 3rd ed. revised and
continued to the end of the 19th century, by A. Hassall, 1901), a
meritorious compilation and storehouse of facts, but not very readable.
The _History of the City of Rome_ (1865) down to the end of the middle
ages was followed by the _History of the Kings of Rome_ (1868), which,
upholding against the German school the general credibility of the
account of early Roman history, given in Livy and other classical
authors, was violently attacked by J.R. Seeley and the _Saturday
Review_, as showing ignorance of the comparative method. More favourable
opinions of the work were expressed by others, but it is generally
agreed that the author's scholarship is defective and that his views are
far too conservative. _Roma Regalis_ (1872) and _A Plea for Livy_ (1873)
were written in reply to his critics. Dyer frequently visited Greece and
Italy, and his topographical works are probably his best; amongst these
mention may be made of _Pompeii, its History, Buildings and Antiquities_
(1867, new ed. in Bohn's _Illustrated Library_), and _Ancient Athens,
its History, Topography and Remains_ (1873). His last publication was
_On Imitative Art_ (1882). He died at Bath on the 30th of January 1888.
DYMOKE, the name of an English family holding the office of king's
champion. The functions of the champion were to ride into Westminster
Hall at the coronation banquet, and challenge all comers to impugn the
king's title (see CHAMPION). The earliest record of the ceremony at the
coronation of an English king dates from the accession of Richard II. On
this occasion the champion was Sir John Dymoke (d. 1381), who held the
manor of Scrivelsby, Lincolnshire, in right of his wife Margaret,
granddaughter of Joan Ludlow, who was the daughter and co-heiress of
Philip Marmion, last Baron Marmion. The Marmions claimed descent from
the lords of Fontenay, hereditary champions of the dukes of Normandy,
and held the castle of Tamworth, Leicestershire, and the manor of
Scrivelsby, Lincolnshire. The right to the championship was disputed
with the Dymoke family by Sir Baldwin de Freville, lord of Tamworth, who
was descended from an elder daughter of Philip Marmion. The court of
claims eventually decided in favour of the owners of Scrivelsby on the
ground that Scrivelsby was held in grand serjeanty, that is, that its
tenure was dependent on rendering a special service, in this case the
championship.
Sir Thomas Dymoke (1428?-1471) joined a Lancastrian rising in 1469, and,
with his brother-in-law Richard, Lord Willoughby and Welles, was
beheaded in 1471 by order of Edward IV. after he had been induced to
leave sanctuary on a promise of personal safety. The estates were
restored to his son Sir Robert Dymoke (d. 1546), champion at the
coronations of Richard III., Henry VII. and Henry VIII., who
distinguished himself at the siege of Tournai and became treasurer of
the kingdom. His descendants acted as champions at successive
coronations. Lewis Dymoke (d. 1820) put in an unsuccessful claim before
the House of Lords for the barony of Marmion. His nephew Henry
(1801-1865) was champion at the coronation of George IV. He was
accompanied on that occasion by the duke of Wellington and Lord Howard
of Effingham. Henry Dymoke was created a baronet; he was succeeded by
his brother John, rector of Scrivelsby (1804-1873), whose son Henry
Lionel died without issue in 1875, when the baronetcy became extinct,
the estate passing to a collateral branch of the family. After the
coronation of George IV. the ceremony was allowed to lapse, but at the
coronation of King Edward VII. H.S. Dymoke bore the standard of England
in Westminster Abbey.
DYNAMICS (from Gr. [Greek: dynamis], strength), the name of a branch of
the science of Mechanics (q.v.). The term was at one time restricted to
the treatment of motion as affected by force, being thus opposed to
Statics, which investigated equilibrium or conditions of rest. In more
recent times the word has been applied comprehensively to the action of
force on bodies either at rest or in motion, thus including "dynamics"
(now termed kinetics) in the restricted sense and "statics."
ANALYTICAL DYNAMICS.--The fundamental principles of dynamics, and their
application to special problems, are explained in the articles MECHANICS
and MOTION, LAWS OF, where brief indications are also given of the more
general methods of investigating the properties of a dynamical system,
independently of the accidents of its particular constitution, which
were inaugurated by J.L. Lagrange. These methods, in addition to the
unity and breadth which they have introduced into the treatment of pure
dynamics, have a peculiar interest in relation to modern physical
speculation, which finds itself confronted in various directions with
the problem of explaining on dynamical principles the properties of
systems whose ultimate mechanism can at present only be vaguely
conjectured. In determining the properties of such systems the methods
of analytical geometry and of the infinitesimal calculus (or, more
generally, of mathematical analysis) are necessarily employed; for this
reason the subject has been named Analytical Dynamics. The following
article is devoted to an outline of such portions of general dynamical
theory as seem to be most important from the physical point of view.
1. _General Equations of Impulsive Motion._
The systems contemplated by Lagrange are composed of discrete
particles, or of rigid bodies, in finite number, connected (it may be)
in various ways by invariable geometrical relations, the fundamental
postulate being that the position of every particle of the system at
any time can be completely specified by means of the instantaneous
values of a finite number of independent variables q1, q2, ... q_n,
each of which admits of continuous variation over a certain range, so
that if x, y, z be the Cartesian co-ordinates of any one particle, we
have for example
x = [f](q1, q2, ... q_n), y = &c., z = &c., (1)
where the functions [f] differ (of course) from particle to particle.
In modern language, the variables q1, q2, ... q_n are _generalized
co-ordinates_ serving to specify the _configuration_ of the system;
their derivatives with respect to the time are denoted by q`1, q`2,
... q`_n, and are called the _generalized components of velocity_. The
continuous sequence of configurations assumed by the system in any
actual or imagined motion (subject to the given connexions) is called
the _path_.
Impulsive motion.
For the purposes of a connected outline of the whole subject it is
convenient to deviate somewhat from the historical order of
development, and to begin with the consideration of _impulsive_
motion. Whatever the actual motion of the system at any instant, we
may conceive it to be generated instantaneously from rest by the
application of proper impulses. On this view we have, if x, y, z be
the rectangular co-ordinates of any particle m,
mx` = X', my` = Y', mz` = Z', (2)
where X', Y', Z' are the components of the impulse on m. Now let
[delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y,
z which are consistent with the connexions of the system, and let us
form the equation
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z), (3)
where the sign [Sigma] indicates (as throughout this article) a
summation extending over all the particles of the system. To transform
(3) into an equation involving the variations [delta]q1, [delta]q2,
... of the generalized co-ordinates, we have
dPx dPx
x` = ---- q`1 + ---- q`2 + ..., &c., &c. (4)
dPq1 dPq2
dPx dPx
[delta]x = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c. (5)
dPq1 dPq2
and therefore
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
(A11q`1 + A12q`2 + ...)[delta]q1 +
(A21q`1 + A22q`2 + ...)[delta]q2 + ..., (6)
where
_ _
| / dPx \ squared / dPy \ squared / dPz \ squared | \
A_rr = [Sigma]m | ( ----- ) + ( ----- ) + ( ----- ) |, |
|_ \dPq_r/ \dPq_r/ \dPq_r/ _| |
_ _ > (7)
| dPx dPx dPy dPy dPz dPz | |
A_rs = [Sigma]m | ----- ----- + ----- ----- + ----- ----- | = A_sr. |
|_dPq_r dPq_s dPq_r dPq_s dPq_r dPq_s _| /
If we form the expression for the kinetic energy [Tau] of the system,
we find
2[Tau] = [Sigma]m(x` squared + y` squared + z` squared) = A11q`1 squared + A22q`2 squared + ...
+ 2A12q`1q`2 + ... (8)
The coefficients A11, A22, ... A12, ... are by an obvious analogy
called the _coefficients of inertia_ of the system; they are in
general functions of the co-ordinates q1, q2, ... . The equation (6)
may now be written
dP[Tau] dP[Tau]
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ------- [delta]q1 + ------- [delta]q2 + ... (9)
dPq`1 dPq`2
This maybe regarded as the cardinal formula in Lagrange's method. For
the right-hand side of (3) we may write
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z) = Q'1[delta]q1 + Q'2[delta]q2 + ... , (10)
where
/ dPx dPy dPz \
Q'_r = [Sigma]( X'----- + Y'----- + Z'----- ). (11)
\ dPq_r dPq_r dPq_r/
The quantities Q1, Q2, ... are called the _generalized components of
impulse_. Comparing (9) and (10), we have, since the variations
[delta]q1, [delta]q2,... are independent,
dP[Tau] dP[Tau]
------- = Q'1, ------- = Q'2, ... (12)
dPq`1 dPq`2
These are the general equations of impulsive motion. It is now usual
to write
dP[Tau]
p_r = ------- (13)
dPq`_r
The quantities p1, p2, ... represent the effects of the several
component impulses on the system, and are therefore called the
_generalized components of momentum_. In terms of them we have
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = p1[delta]q1 + p2[delta]q2 + ... (14)
Also, since [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2 ...,
2[Tau] = p1q`1 + p2q`2 + ... (15)
This follows independently from (14), assuming the special variations
[delta]x = x`dt, &c., and therefore [delta]q1 = q`1dt, [delta]q2 =
q`2dt, ...
Reciprocal theorems.
Again, if the values of the velocities and the momenta in any other
motion of the system through the same configuration be distinguished
by accents, we have the identity
p1q`'1 + p2q`'2 + ... = p'1q`1 + p'2q`2 + ..., (16)
each side being equal to the symmetrical expression
A11q`1q''1 + A22q`2q`'2 + ... + A12(q`1q`'2 + q`'1q`2) + ... (17)
The theorem (16) leads to some important reciprocal relations. Thus,
let us suppose that the momenta p1, p2, ... all vanish with the
exception of p1, and similarly that the momenta p'1, p'2, ... all
vanish except p'2. We have then p1q`'1 = p'2q`2, or
q`2 : p1 = q`'1 : p'2 (18)
The interpretation is simplest when the co-ordinates q1, q2 are both
of the same kind, e.g. both lines or both angles. We may then
conveniently put p1 = p'2, and assert that the velocity of the first
type due to an impulse of the second type is equal to the velocity of
the second type due to an equal impulse of the first type. As an
example, suppose we have a chain of straight links hinged each to the
next, extended in a straight line, and free to move. A blow at right
angles to the chain, at any point P, will produce a certain velocity
at any other point Q; the theorem asserts that an equal velocity will
be produced at P by an equal blow at Q. Again, an impulsive couple
acting on any link A will produce a certain angular velocity in any
other link B; an equal couple applied to B will produce an equal
angular velocity in A. Also if an impulse F applied at P produce an
angular velocity [omega] in a link A, a couple Fa applied to A will
produce a linear velocity [omega]a at P. Historically, we may note
that reciprocal relations in dynamics were first recognized by H.L.F.
Helmholtz in the domain of acoustics; their use has been greatly
extended by Lord Rayleigh.
Velocities in terms of momenta.
The equations (13) determine the momenta p1, p2,... as linear
functions of the velocities q`1, q`2,... Solving these, we can
express q`1, q`2 ... as linear functions of p1, p2,... The
resulting equations give us the velocities produced by any given
system of impulses. Further, by substitution in (8), we can express
the kinetic energy as a homogeneous quadratic function of the momenta
p1, p2,... The kinetic energy, _as so expressed_, will be denoted by
[Tau]'; thus
2[Tau]' = A'11p1 squared + A'22p2 squared + ... + 2A'12p - p2 + ... (19)
where A'11, A'22,... A'12,... are certain coefficients depending on
the configuration. They have been called by Maxwell the _coefficients
of mobility_ of the system. When the form (19) is given, the values
of the velocities in terms of the momenta can be expressed in a
remarkable form due to Sir W.R. Hamilton. The formula (15) may be
written
p1q`1 + p2q`2 + ... = [Tau] + [Tau]', ... (20)
where [Tau] is supposed expressed as in (8), and [Tau]' as in (19).
Hence if, for the moment, we denote by [delta] a variation affecting
the velocities, and therefore the momenta, but not the configuration,
we have
p1[delta]q`1 + q`1[delta]p + p2[delta]q`2 + q`2[delta]p2 + ... = [delta][Tau] + [delta][Tau]'
dP[Tau] dP[Tau] dP[Tau]' dP[Tau]'
= ------- [delta]q`1 + ------- [delta]q`2 + ... + -------- [delta]p1 + -------- [delta]p2 + ... (21)
dPq`1 dPq`2 dPp1 dPp2
In virtue of (13) this reduces to
dP[Tau]' dP[Tau]'
q`1[delta]p1 + q`2[delta]p2 + ... = ------- [delta]p1 + ------- [delta]p2 + ... (22)
dPp1 dPp2
Since [delta]p1, [delta]p2, ... may be taken to be independent, we
infer that
dP[Tau]' dP[Tau]'
q`1 = -------, q`2 = -------, ... (23)
dPp1 dPp2
In the very remarkable exposition of the matter given by James Clerk
Maxwell in his _Electricity and Magnetism_, the Hamiltonian
expressions (23) for the velocities in terms of the impulses are
obtained directly from first principles, and the formulae (13) are
then deduced by an inversion of the above argument.
Routh's modification.
An important modification of the above process was introduced by E.J.
Routh and Lord Kelvin and P.G. Tait. Instead of expressing the kinetic
energy in terms of the velocities alone, or in terms of the momenta
alone, we may express it in terms of the velocities corresponding to
some of the co-ordinates, say q1, q2, ... q_m, and of the momenta
corresponding to the remaining co-ordinates, which (for the sake of
distinction) we may denote by [chi], [chi]', [chi]", .... Thus, [Tau]
being expressed as a homogeneous quadratic function of q`1, q`2, ...
q`_m, [chi]`, [chi]`', [chi]`", ..., the momenta corresponding to the
co-ordinates [chi], [chi]', [chi]", ... may be written
dP[Tau] dP[Tau] dP[Tau]
[kappa] = --------, [kappa]' = ---------, [kappa]" = ------------, ... (24)
dP[chi]` dP[chi]`' dP[.[chi]`"
These equations, when written out in full, determine [chi]`, [chi]`',
[chi]`", ... as linear functions of q`1, q`2, ... q`_m, [kappa],
[kappa]', [kappa]",... We now consider the function
R = [Tau] - [kappa][chi]' - [kappa]'[chi]]`' - [kappa]"[chi]]`" - ..., (25)
supposed expressed, by means of the above relations in terms of q`1,
q`2, ... q`_m, [kappa], [kappa]', [kappa]",... Performing the
operation [delta] on both sides of (25), we have
dPR dPR dP[Tau] dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... + -------- [delta][chi]` + ...
dPq`1 dP[kappa] dPq`1 dP[chi]`
- [kappa]dP[chi]` - [chi]`[delta][kappa] - ... , (26)
where, for brevity, only one term of each type has been exhibited.
Omitting the terms which cancel in virtue of (24), we have
dPR dPR dP[Tau]
----- [delta]q`1 + ... + --------- [delta][kappa] + ... = ------- [delta]q`1 + ... - [chi]`[delta][kappa] - ... (27)
dPq`1 dP[kappa] dPq`1
Since the variations [delta]q1, [delta]q2, ... [delta]q_m,
[delta][kappa], [delta][kappa]', [delta][kappa]", ... may be taken to
be independent, we have
dP[Tau] dPR dP[Tau] dPR
p1 = ------- = -----, p2 = ------- = -----, ... (28)
dPq`1 dPq`1 dPq`2 dPq`2
and
dPR dPR dPR
[chi]` = - ---------, [chi]`' = - ----------, [chi]]`" = - ---------, ... (29)
dP[kappa] dP[kappa]' dP[kappa]"
An important property of the present transformation is that, when
expressed in terms of the new variables, the kinetic energy is the sum
of two homogeneous quadratic functions, thus
[Tau] = [@] + K, (30)
where [@] involves the velocities q`1, q`2, ... q`_m alone, and K the
momenta [kappa], [kappa]', [kappa]", ... alone. For in virtue of (29)
we have, from (25),
/ dPR dPR dPR \
[Tau] = R - ( [kappa] --------- + [kappa]' ---------- + [kappa]" ----------- + ... ), (31)
\ dP[kappa] dP[kappa]' dP[kappa]" /
and it is evident that the terms in R which are bilinear in respect of
the two sets of variables q`1, q`2, ... q`_m and [kappa], [kappa]',
[kappa]", ... will disappear from the right-hand side.
Maximum and minimum energy.
It may be noted that the formula (30) gives immediate proof of two
important theorems due to Bertrand and to Lord Kelvin respectively.
Let us suppose, in the first place, that the system is started by
given impulses of certain types, but is otherwise free. J.L.F.
Bertrand's theorem is to the effect that the kinetic energy is
_greater_ than if by impulses of the remaining types the system were
constrained to take any other course. We may suppose the co-ordinates
to be so chosen that the constraint is expressed by the vanishing of
the velocities q`1, q`2, ... q`_m, whilst the given impulses are
[kappa], [kappa]', [kappa]",... Hence the energy in the actual motion
is greater than in the constrained motion by the amount [@].
Again, suppose that the system is started with prescribed velocity
components q`1, q`2, ... q`_m, by means of proper impulses of
the corresponding types, but is otherwise free, so that in the motion
actually generated we have [kappa] = 0, [kappa]' = 0, [kappa]" = 0,
... and therefore K = 0. The kinetic energy is therefore _less_ than
in any other motion consistent with the prescribed velocity-conditions
by the value which K assumes when [kappa], [kappa]', [kappa]", ...
represent the impulses due to the constraints.
Simple illustrations of these theorems are afforded by the chain of
straight links already employed. Thus if a point of the chain be held
fixed, or if one or more of the joints be made rigid, the energy
generated by any given impulses is less than if the chain had
possessed its former freedom.
2. _Continuous Motion of a System._
Lagrange's equations.
We may proceed to the continuous motion of a system. The equations of
motion of any particle of the system are of the form
mx" = X, my" = Y, mz" = Z (1)
Now let x + [delta]x, y + [delta]y, z + [delta]z be the co-ordinates
of m in any arbitrary motion of the system differing infinitely little
from the actual motion, and let us form the equation
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) =
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) (2)
Lagrange's investigation consists in the transformation of (2) into an
equation involving the independent variations [delta]q1, [delta]q2,
... [delta]q_n.
It is important to notice that the symbols [delta] and d/dt are
commutative, since
d dx d
[delta]x` = --(x + [delta]x) - -- = --[delta]x, &c. (3)
dt dt dt
Hence
d
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = -- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)
dt
- [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)
d
= --(p1[delta]q1 + p2[delta]q2 + ...) - [delta][Tau], (4)
dt
by Sec. 1 (14). The last member may be written
p`1[delta]q1 + p1[delta]q`1 + p`2[delta]q2 + p2[delta]q`2 + ...
dP[Tau] dP[Tau] dP[Tau] dP[Tau]
- ------- [delta]q`1 - ------- [delta]q1 - ------- [delta]q`2 - ------- [delta]q2 - ... (5)
dPq`1 dPq1 dPq`2 dPq2
Hence, omitting the terms which cancel in virtue of Sec. 1 (13), we find
/ dP[Tau]\ / dP[Tau]\
[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) = (p`1 - ------- ) [delta]q1 + (p`2 - ------- ) [delta]q2 + ... (6)
\ dPq1 / \ dPq2 /
For the right-hand side of (2) we have
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = Q1[delta]q1 + Q2[delta]q2 + ..., (7)
/ dPx dPy dPz \
where Q_r = [Sigma]( X ----- + Y ----- + Z ----- ) (8)
\ dPq_r dPq_r dPq_r/
The quantities Q1, Q2, ... are called the _generalized components of
force_ acting on the system.
Comparing (6) and (7) we find
dP[Tau] dP[Tau]
p`1 - ------- = Q1, p`2 - ------- = Q2, ..., (9)
dPq`1 dPq`2
or, restoring the values of p1, p2, ...,
d /dP[Tau]\ dP[Tau] d /dP[Tau]\ dP[Tau]
-- ( ------- ) - ------- = Q1, -- ( ------- ) - ------- = Q2, ... (10)
dt \ dPq`1 / dPq1 dt \ dPq`2 / dPq2
These are Lagrange's general equations of motion. Their number is of
course equal to that of the co-ordinates q1, q2, ... to be determined.
Analytically, the above proof is that given by Lagrange, but the
terminology employed is of much more recent date, having been first
introduced by Lord Kelvin and P.G. Tait; it has greatly promoted the
physical application of the subject. Another proof of the equations
(10), by direct transformation of co-ordinates, has been given by
Hamilton and independently by other writers (see MECHANICS), but the
variational method of Lagrange is that which stands in closest
relation to the subsequent developments of the subject. The chapter of
Maxwell, already referred to, is a most instructive commentary on the
subject from the physical point of view, although the proof there
attempted of the equations (10) is fallacious.
In a "conservative system" the work which would have to be done by
extraneous forces to bring the system from rest in some standard
configuration to rest in the configuration (q1, q2, ... q_n) is
independent of the path, and may therefore be regarded as a definite
function of q1, q2, ... q_n. Denoting this function (the _potential
energy_) by V, we have, if there be no extraneous force on the system,
[Sigma](X[delta]x + Y[delta]y + Z[delta]z) = - [delta]V, (11)
and therefore
dPV dPV
Q1 = - ----, Q2 = - ----, .... (12)
dPq1 dPq2
Hence the typical Lagrange's equation may be now written in the form
d /dP[Tau]\ dP[Tau] dPV
-- ( ------- ) - ------- = - -----, (13)
dt \dPq`_r / dPq_r dPq_r
or, again,
dP
p`_r = - ----- (V - [Tau]) (14)
dPq_r
It has been proposed by Helmholtz to give the name _kinetic potential_
to the combination V - [Tau].
As shown under MECHANICS, Sec. 22, we derive from (10)
d[Tau]
------ = Q1q`1 + Q2q`2 + ..., (15)
dt
and therefore in the case of a conservative system free from
extraneous force,
d
--([Tau] + V) = 0 or [Tau] + V = const., (16)
dt
which is the equation of energy. For examples of the application of
the formula (13) see MECHANICS, Sec. 22.
3. _Constrained Systems._
Case of varying relations.
It has so far been assumed that the geometrical relations, if any,
which exist between the various parts of the system are of the type Sec.
1 (1), and so do not contain t explicitly. The extension of Lagrange's
equations to the case of "varying relations" of the type
x = f(t, q1, q2,...q_n), y = &c., z = &c., (1)
was made by J.M.L. Vieille. We now have
dPx dPx dPx
x` = --- + ---- q`1 + ---- q`2 + ..., &c., &c., (2)
dPt dPq1 dPq2
dPx dPx
dPx = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c., (3)
dPq1 dPq2
so that the expression Sec. 1 (8) for the kinetic energy is to be
replaced by
2[Tau] = [alpha]0 + 2[alpha]1q`1 + 2[alpha]2q`2 + ... + A11q`1 squared + A22q`2 squared + ... + A12q`1q`2 + ..., (4)
where
_ _ \
| /dPx\ squared /dPy\ squared /dPz\ squared | |
a0 = [Sigma]m |( --- ) + ( --- ) + ( --- ) |, |
|_\dPt/ \dPt/ \dPt/ _| |
_ _ > (5)
| dPx dPx dPy dPy dPz dPz | |
a_r = [Sigma]m | --- ----- + --- ----- + --- ----- |, |
|_dPt dPq_r dPt dPq_r dPt dPq_r_| |
/
and the forms of A_rr, A_rs are as given by Sec. 1 (7). It is to be
remembered that the coefficients [alpha]0, [alpha]1, [alpha]2, ...
A11, A22, ... A12 ... will in general involve t explicitly as well as
implicitly through the co-ordinates q1, q2,... Again, we find
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
([alpha]1 + A11q`1 + A12q`2 + ...)[delta]q1
+ ([alpha]2 + A21q`1 + A22q`2 + ...)dPq2 + ...
dP[Tau] dP[Tau]
= ------- [delta]q1 + ------- [delta]q2 + ...
dPq`1 dPq`2
= p1[delta]q1 + p2[delta]q2 + ..., (6)
where p_r is defined as in Sec. 1 (13). The derivation of Lagrange's
equations then follows exactly as before. It is to be noted that the
equation Sec. 2 (15) does not as a rule now hold. The proof involved the
assumption that [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2....
It has been pointed out by R.B. Hayward that Vieille's case can be
brought under Lagrange's by introducing a new co-ordinate ([chi]) in
place of t, so far as it appears explicitly in the relations (1). We
have then
2[Tau] = [alpha]0[chi]` squared + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ A11q`1 squared + A22q`2 squared + ... + 2A12q`1q`2 + .... (7)
The equations of motion will be as in Sec. 2 (10), with the additional
equation
d dP[Tau] dP[Tau]
-- -------- - ------- = X, (8)
dt dP[chi]` dP[chi]
where X is the force corresponding to the co-ordinate [chi]. We may
suppose X to be adjusted so as to make [chi]" = 0, and in the
remaining equations nothing is altered if we write t for [chi] before,
instead of after, the differentiations. The reason why the equation Sec.
2 (15) no longer holds is that we should require to add a term X[chi]`
on the right-hand side; this represents the rate at which work is
being done by the constraining forces required to keep [chi]`
constant.
As an example, let x, y, z be the co-ordinates of a particle relative
to axes fixed in a solid which is free to rotate about the axis of z.
If [phi] be the angular co-ordinate of the solid, we find without
difficulty
2[Tau] = m(x` squared + y` squared +z` squared) + 2[phi]`m(xy` - yx`) + {I + m(x squared + y squared)}[phi]` squared, (9)
where I is the moment of inertia of the solid. The equations of
motion, viz.
d dP[Tau] dP[Tau] d dP[Tau] dP[Tau] d dP[Tau] dP[Tau]
-- ------ - ------ = X, -- ------- - ------- = Y, -- ------- - ------- = Z, (10)
dt dPx` dPx dt dPy` dPy dt dPz` dPz
d dP[Tau] dP[Tau]
and -- -------- - ------- = [Phi], (11)
dt dP[phi]` dP[phi]
become
m(x" - 2[phi]`y` - x[phi]` squared - y[phi]") = X, m(y" + 2[phi]`x` - y[phi]` squared + x[phi]`) = Y, mz" = Z, (12)
_ _
d | / \ |
and -- |(I + m(x squared + y squared)) [phi]` + m(xy` - yx`)| = [Phi]. (13)
dt |_\ / _|
If we suppose [Phi] adjusted so as to maintain [phi]" = 0, or (again)
if we suppose the moment of inertia I to be infinitely great, we
obtain the familiar equations of motion relative to moving axes, viz.
m(x" - 2[omega]y` - [omega] squaredx) = X, m(y" + 2[omega]x` - [omega] squaredy) = Y, mz" = Z, (14)
where [omega] has been written for [phi]. These are the equations
which we should have obtained by applying Lagrange's rule at once to
the formula
2[Tau] = m(x` squared + y` squared + z` squared) + 2m[omega](xy` - yx`) + m[omega] squared(x squared + y squared), (15)
which gives the kinetic energy of the particle referred to axes
rotating with the constant angular velocity [omega]. (See MECHANICS, Sec.
13.)
More generally, let us suppose that we have a certain group of
co-ordinates [chi], [chi]', [chi]", ... whose absolute values do not
affect the expression for the kinetic energy, and that by suitable
forces of the corresponding types the velocity-components [chi]`,
[chi]`', [chi]`", ... are maintained constant. The remaining
co-ordinates being denoted by q1, q2, ... q_n, we may write
2[Tau] = [@] + [Tau]0 + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ 2([alpha]'1q`1 + [alpha]'2q`2 + ...)[chi]`' + ..., (16)
where [@] is a homogeneous quadratic function of the velocities q`1,
q`2, ... q`_n of the type Sec.1 (8), whilst [Tau]0 is a homogeneous
quadratic function of the velocities [chi]`,[chi]`', [chi]`", ...
alone. The remaining terms, which are bilinear in respect of the two
sets of velocities, are indicated more fully. The formulae (10) of Sec. 2
give n equations of the type
d /dP[@]\ /dP[@]\ dP[Tau]0
--( ----- ) - ( ----- ) + (r, 1)q`1 + (r, 2)q`2 + ... - -------- = Q_r (17)
dt \dPq_r/ \dPq_r/ dPq_r
where
/dPa_r dPa_s\ /dPa'_r dPa'_s\
(r, s) = ( ----- - ----- )[chi]` + ( ------ - ------ )[chi]`' + .... (18)
\dPq_s dPq_r/ \dPq_s dPq_r/
These quantities (r, s) are subject to the relations
(r, s) = -(s, r), (r, r) = 0 (19)
The remaining dynamical equations, equal in number to the co-ordinates
[chi], [chi]', [chi]", ..., yield expressions for the forces which
must be applied in order to maintain the velocities [chi]`, [chi]`',
[chi]`", ... constant; they need not be written down. If we follow the
method by which the equation of energy was established in Sec. 2, the
equations (17) lead, on taking account of the relations (19), to
d
--([@] - [Tau]0) = Q1q`1 + Q2q`2 + ... + Q_nq`_n, (20)
dt
or, in case the forces Q_r depend only on the co-ordinates q1, q2, ...
q_n and are conservative,
[@] + V - [Tau]0 = const. (21)
The conditions that the equations (17) should be satisfied by zero
values of the velocities q`1, q`2, ... q`_n are
dP[Tau]0
Q_r = - --------, (22)
dPq_r
or in the case of conservative forces
dP
------ (V - [Tau]0) = 0, (23)
dPq_r
i.e. the value of V - [Tau]0 must be _stationary_.
Rotating axes.
We may apply this to the case of a system whose configuration relative
to axes rotating with constant angular velocity ([omega]) is defined
by means of the n co-ordinates q1, q2, ... q_n. This is important on
account of its bearing on the kinetic theory of the tides. Since the
Cartesian co-ordinates x, y, z of any particle m of the system
relative to the moving axes are functions of q1, q2, ... q_n, of the
form Sec. 1 (1), we have, by (15)
2[@] = [Sigma]m(x` squared + y` squared + z` squared), 2[Tau]0 = [omega] squared[Sigma]m(x squared + y squared), (24)
/ dPy dPx \
a_r= [Sigma]m( x----- - y----- ), (25)
\ dPq_r dPq_r/
whence
dP(x, y)
(r, s) = 2[omega].[Sigma]m ------------. (26)
dP(q_s, q_r)
The conditions of relative equilibrium are given by (23).
It will be noticed that this expression V - [Tau]0, which is to be
stationary, differs from the true potential energy by a term which
represents the potential energy of the system in relation to
fictitious "centrifugal forces." The question of stability of relative
equilibrium will be noticed later (Sec. 6).
It should be observed that the remarkable formula (20) may in the
present case be obtained directly as follows. From (15) and (14) we
find
d[Tau] d
------ = --([@] + [Tau]0) + [omega].[Sigma]m(xy" - yx")
dt dt
d
= --([@] - [Tau]0) + [omega].[Sigma](xY - yX). (27)
dt
This must be equal to the rate at which the forces acting on the
system do work, viz. to
[omega][Sigma](xY - yX) + Q1q`1 + Q2q`2 + ... + Q_nq`_n,
where the first term represents the work done in virtue of the
rotation.
Constrained systems.
We have still to notice the modifications which Lagrange's equations
undergo when the co-ordinates q1, q2, ... q_n are not all
independently variable. In the first place, we may suppose them
connected by a number m ( < n) of relations of the type
A(t, q1, q2, ... q_n) = 0, B(t, q1, q2, ... q_n) = 0, &c. (28)
These may be interpreted as introducing partial constraints into a
previously free system. The variations [delta]q1, [delta]q2, ...
[delta]q_n in the expressions (6) and (7) of Sec. 2 which are to be
equated are no longer independent, but are subject to the relations
dPA dPA dPB dPB
---- [delta]q1 + ---- [delta]q2 + ... = 0, ---- [delta]q1 + ---- [delta]q2 + ... = 0, &c. (29)
dPq1 dPq2 dPq1 dPq2
Introducing indeterminate multipliers [lambda], mu, ..., one for each
of these equations, we obtain in the usual manner n equations of the
type
d dP[Tau] dP[Tau] dPA dPB
-- ------- - ------- = Q_r + [lambda] ----- + mu ----- + ..., (30)
dt dPq`_r dPq_r dPq_r dPq_r
in place of Sec. 2 (10). These equations, together with (28), serve to
determine the n co-ordinates q1, q2, ... q_n and the m multipliers
[lambda], mu, ....
When t does not occur explicitly in the relations (28) the system is
said to be _holonomic_. The term connotes the existence of integral
(as opposed to differential) relations between the co-ordinates,
independent of the time.
Again, it may happen that although there are no prescribed relations
between the co-ordinates q1, q2, ... q_n, yet from the circumstances
of the problem certain geometrical conditions are imposed on their
_variations_, thus
A1[delta]q1 + A2[delta]q2 + ... = 0, B1[delta]q1 + B2[delta]q2 + ... = 0, &c., (31)
where the coefficients are functions of q1, q2, ... q_n and (possibly)
of t. It is assumed that these equations are not integrable as regards
the variables q1, q2, ... q_n; otherwise, we fall back on the previous
conditions. Cases of the present type arise, for instance, in ordinary
dynamics when we have a solid rolling on a (fixed or moving) surface.
The six co-ordinates which serve to specify the position of the solid
at any instant are not subject to any necessary relation, but the
conditions to be satisfied at the point of contact impose three
conditions of the form (31). The general equations of motion are
obtained, as before, by the method of indeterminate multipliers, thus
d dP[Tau] dP[Tau]
-- ------- - ------- = Q_r + [lambda]A_r + muB_r + ... (32)
dt dPq`_r dPq_r
The co-ordinates q1, q2, ... q_n, and the indeterminate multipliers
[lambda], mu, ..., are determined by these equations and by the
velocity-conditions corresponding to (31). When t does not appear
explicitly in the coefficients, these velocity-conditions take the
forms
A1q`1 + A2q`2 + ... = 0, B1q`1 + B2q`2 + ... = 0, &c. (33)
Systems of this kind, where the relations (31) are not integrable, are
called _non-holonomic_.
4. _Hamiltonian Equations of Motion._
In the Hamiltonian form of the equations of motion of a conservative
system with unvarying relations, the kinetic energy is supposed
expressed in terms of the _momenta_ p1, p2, ... and the co-ordinates
q1, q2, ..., as in Sec. 1 (19). Since the symbol [delta] now denotes a
variation extending to the co-ordinates as well as to the momenta, we
must add to the last member of Sec. 1 (21) terms of the types
dP[Tau] dP[Tau]'
------- [delta]q1 + -------- [delta]q1 + ... (1)
dPq1 dPq1
Since the variations [delta]p1, [delta]p2, ... [delta]q1, [delta]q2,
... may be taken to be independent, we infer the equations Sec. 1 (23) as
before, together with
dP[Tau] dP[Tau]' dP[Tau] dP[Tau]'
------ = - --------, ------- = - --------, ..., (2)
dPq1 dPq1 dPq2 dPq2
Hence the Lagrangian equations Sec. 2 (14) transform into
dP dP
p`1 = - ----([Tau]' + V), p`2 = ---- ([Tau]' + V), ... (3)
dPq1 dPq2
If we write
H = [Tau]' + V, (4)
so that H denotes the _total energy_ of the system, supposed expressed
in terms of the new variables, we get
dPH dPH
p`1 = - ----, p`2 = - ----, ... (5)
dPq1 dPq2
If to these we join the equations
dPH dPH
q`1 = ----, q`2 = ----, ..., (6)
dPp1 dPp2
which follow at once from Sec. 1 (23), since V does not involve p1, p2,
..., we obtain a complete system of differential equations _of the
first order_ for the determination of the motion.
The equation of energy is verified immediately by (5) and (6), since
these make
dH dPH dPH dPH dPH
-- = ---- p`1 + ---- p`2 + ... + ---- q`1 + ---- q`2 + ... = 0. (7)
dt dPp1 dPp2 dPq1 dPq2
The Hamiltonian transformation is extended to the case of varying
relations as follows. Instead of (4) we write
H = p1q`1 + p2q`2 + ... - [Tau] + V, (8)
and imagine H to be expressed in terms of the momenta p1, p2, ..., the
co-ordinates q1, q2, ..., and the time. The internal forces of the
system are assumed to be conservative, with the potential energy V.
Performing the variation [delta] on both sides, we find
dP[Tau] dPV
[delta]H = q`1[delta]p1 + ... - ------- [delta]q1 + ---- [delta]q + ..., (9)
dPq1 dPq1
terms which cancel in virtue of the definition of p1, p2, ... being
omitted. Since [delta]p1, [delta]p2, ..., [delta]q1, [delta]q2, ...
may be taken to be independent, we infer
dPH dPH
q`1 = ----, q`2 = ----, ..., (10)
dPp1 dPp2
and
dP dPH dP dPH
---- ([Tau] - V) = - ----, ----([Tau] - V) = - ----, .... (11)
dPq1 dPq1 dPq2 dPq2
It follows from (11) that
dPH dPH
p`1 = - ----, p`2 = - ----, .... (12)
dPq1 dPq2
The equations (10) and (12) have the same form as above, but H is no
longer equal to the energy of the system.
5. _Cyclic Systems._
A _cyclic_ or _gyrostatic_ system is characterized by the following
properties. In the first place, the kinetic energy is not affected if
we alter the absolute values of certain of the co-ordinates, which we
will denote by [chi], [chi]', [chi]", ..., provided the remaining
co-ordinates q1, q2, ... q_m and the velocities, including of course
the velocities [.[chi]], [.[chi]]', [.[chi]]", ..., are unaltered.
Secondly, there are no forces acting on the system of the types [chi],
[chi]', [chi]", .... This case arises, for example, when the system
includes gyrostats which are free to rotate about their axes, the
co-ordinates [chi], [chi]', [chi]", ... then being the angular
co-ordinates of the gyrostats relatively to their frames. Again, in
theoretical hydrodynamics we have the problem of moving solids in a
frictionless liquid; the ignored co-ordinates [chi], [chi]', [chi]",
... then refer to the fluid, and are infinite in number. The same
question presents itself in various physical speculations where
certain phenomena are ascribed to the existence of _latent motions_ in
the ultimate constituents of matter. The general theory of such
systems has been treated by E.J. Routh, Lord Kelvin, and H.L.F.
Helmholtz.
Routh's equations.
If we suppose the kinetic energy [Tau] to be expressed, as in
Lagrange's method, in terms of the co-ordinates and the velocities,
the equations of motion corresponding to [chi], [chi]', [chi]'', ...
reduce, in virtue of the above hypotheses, to the forms
d dP[Tau] d dP[Tau] d dP[Tau]
-- --------- = 0, -- --------- = 0, -- --------- = 0, ..., (1)
dt dP[chi]` dt dP[chi]`' dt dP[chi]`"
whence
dP[Tau] dP[Tau] dP[Tau]
-------- = [kappa], --------- = [kappa]', --------- = [kappa]", ..., (2)
dP[chi]` dP[chi]`' dP[chi]`"
where [kappa], [kappa]', [kappa]", ... are the constant momenta
corresponding to the cyclic co-ordinates [chi], [chi]', [chi]", ....
These equations are linear in [.[chi]], [.[chi]]', [.[chi]]", ...;
solving them with respect to these quantities and substituting in the
remaining Lagrangian equations, we obtain m differential equations to
determine the remaining co-ordinates q1, q2, ... q_m. The object of
the present investigation is to ascertain the general form of the
resulting equations. The retained co-ordinates q1, q2, ... q_m may be
called (for distinction) the _palpable_ co-ordinates of the system; in
many practical questions they are the only co-ordinates directly in
evidence.
If, as in Sec. 1 (25), we write
R = [Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ..., (3)
and imagine R to be expressed by means of (2) as a quadratic function
of q`1, q`2, ... q`_m, [kappa], [kappa]', [kappa]", ... with
coefficients which are in general functions of the co-ordinates q1,
q2, ... q_m, then, performing the operation [delta] on both sides, we
find
dPR dPR dPR dP[Tau] dP[Tau]
-----[delta]q`1 + ... + ---------[delta][kappa] + ... + ----[delta]q1 + ... = -------[delta]q`1 + ... + -------[delta]q1 + ...
dPq`1 dP[kappa] dPq1 dPq`1 dPq1
dP[Tau] dP[Tau]
+ --------[delta][chi]` + ... + --------[delta]q1 + ... - [kappa][delta][chi]` - [chi]`[delta][kappa] - .... (4)
dP[chi]` dP[chi]1
Omitting the terms which cancel by (2), we find
dP[Tau] dPR dP[Tau] dPR
------- = -----, ------- = -----, ..., (5)
dPq`1 dPq`1 dPq`2 dPq`2
dP[Tau] dPR dP[Tau] dPR
------- = ----, ------- = ----, ..., (6)
dPq1 dPq1 dPq2 dPq2
dPR dPR dPR
[chi]` = - ---------, [chi]`' = - ----------, [chi]`" = - ----------, ... (7)
dP[kappa] dP[kappa]' dP[kappa]"
Substituting in Sec. 2 (10), we have
d dPR dPR d dPR dPR
-- ----- - ----- = Q1, -- ----- - ---- = Q2, ... (8)
dt dPq`1 dPq1 dt dPq`2 dPq2
These are Routh's forms of the modified Lagrangian equations.
Equivalent forms were obtained independently by Helmholtz at a later
date.
Kelvin's equations.
The function R is made up of three parts, thus
R = R(2,0) + R(1,1) + R(0,2), ... (9)
where R(2,0) is a homogeneous quadratic function of q`1, q`2, ...
q`_m, R(0,2) is a homogeneous quadratic function of [kappa], [kappa]',
[kappa]", ..., whilst R(1,1) consists of products of the velocities
q`1, q`2, ... q`_m into the momenta [kappa], [kappa]', [kappa]"....
Hence from (3) and (7) we have
/ dPR dPR dPR \
[Tau] = R - ( [kappa] --------- + [kappa]'---------- + [kappa]" ---------- + ...)
\ dP[kappa] dP[kappa]' dP[kappa]" /
= R(2,0) - R(0,2). (10)
If, as in Sec. 1 (30), we write this in the form
[Tau] = [@] + [Kappa], (11)
then (3) may be written
R = [@] - [Kappa] + ss1q`1 + ss2q`2 + ..., (12)
where ss1, ss2, ... are linear functions of [kappa], [kappa]', [kappa]",
..., say
ss_r = [alpha]_r[kappa] + [alpha]'_r[kappa]' + [alpha]"_r[kappa]" + ..., (13)
the coefficients [alpha]_r, [alpha]'_r, [alpha]"_r, ... being in
general functions of the co-ordinates q1, q2, ... q_m. Evidently ss_r
denotes that part of the momentum-component dPR/dPq`_r which is due to
the cyclic motions. Now
d dPR d / dP[@] \ d dP[@] dPss_r dPss_r
-- ------ = -- ( ------ + ss_r) = -- ------ + -----q`1 + -----q`2 + ..., (14)
dt dPq`_r dt \dPq`_r / dt dPq`_r dPq1 dPq2
dPR dP[@] dP[Kappa] dPss1 dPss2
----- = ----- - --------- + -----q`1 + -----q`2 + .... (15)
dPq_r dPq_r dPq_r dPq_r dPq_r
Hence, substituting in (8), we obtain the typical equation of motion
of a gyrostatic system in the form
d dP[@] dP[@] dP[Kappa]
-- ------ - ----- + (r, 1)q`1 + (r, 2)q`2 + ... + (r, s)q`_s + ... + --------- = Q_r, (16)
dt dPq`_r dPq_r dPq_r
where
dPss_r dPss_s
(r, s) = ----- - -----. (17)
dPq_s dPq_r
This form is due to Lord Kelvin. When q1, q2, ... q_m have been
determined, as functions of the time, the velocities corresponding to
the cyclic co-ordinates can be found, if required, from the relations
(7), which may be written
dP[Kappa] \
[Chi]` = --------- - [alpha]1q`1 - [alpha]2q`2 - ..., |
dP[kappa] |
|
dP[Kappa] > (18)
[Chi]`' = ---------- - [alpha]'1q`1 - [alpha]'2q`2 - ..., |
dP[kappa]' |
|
&c., &c. /
It is to be particularly noticed that
(r, r) = 0, (r, s) = -(s, r). (19)
Hence, if in (16) we put r = 1, 2, 3, ... m, and multiply by q`1,
q`2, ... q`_m respectively, and add, we find
d
--([@] + [Kappa]) = Q1q`1 + Q2q`2 + ..., (20)
dt
or, in the case of a conservative system
[@] + V + [Kappa] = const., (21)
which is the equation of energy.
The equation (16) includes Sec. 3 (17) as a particular case, the
eliminated co-ordinate being the angular co-ordinate of a rotating
solid having an infinite moment of inertia.
In the particular case where the cyclic momenta [kappa], [kappa]',
[kappa]", ... are all zero, (16) reduces to
d dP[@] dP[@]
-- ------ - ----- = Q_r. (22)
dt dPq`_r dPq_r
The form is the same as in Sec. 2, and the system now behaves, as regards
the co-ordinates q1, q2, ... q_m, exactly like the acyclic type there
contemplated. These co-ordinates do not, however, now fix the position
of every particle of the system. For example, if by suitable forces
the system be brought back to its initial configuration (so far as
this is defined by q1, q2, ..., q_m), after performing any evolutions,
the ignored co-ordinates [chi], [chi]', [chi]", ... will not in
general return to their original values.
If in Lagrange's equations Sec. 2 (10) we reverse the sign of the
time-element dt, the equations are unaltered. The motion is therefore
reversible; that is to say, if as the system is passing through any
configuration its velocities q`1, q`2, ..., q`_m be all
reversed, it will (if the forces be the same in the same
configuration) retrace its former path. But it is important to observe
that the statement does not in general hold of a gyrostatic system;
the terms of (16), which are linear in q`1, q`2, ..., q`_m,
change sign with dt, whilst the others do not. Hence the motion of a
gyrostatic system is not reversible, unless indeed we reverse the
cyclic motions as well as the velocities q`1, q`2, ..., q`_m.
For instance, the precessional motion of a top cannot be reversed
unless we reverse the spin.
Kinetostatics.
The _conditions of equilibrium_ of a system with latent cyclic motions
are obtained by putting q`1 = 0, q`2 = 0, ... q`_m = 0 in (16); viz.
they are
dP[Kappa] dP[Kappa]
Q1 = ---------, Q2 = ---------, ... (23)
dPq1 dPq2
These may of course be obtained independently. Thus if the system be
guided from (apparent) rest in the configuration (q1, q2, ... q_m) to
rest in the configuration q1 + [delta]q1, q2 + [delta]q2, ..., q_m +
[delta]q_m, the work done by the forces must be equal to the increment
of the kinetic energy. Hence
Q1[delta]q1 + Q2[delta]q2 + ... = [delta][Kappa], (24)
which is equivalent to (23). The conditions are the same as for the
equilibrium of a system without latent motion, but endowed with
potential energy [Kappa]. This is important from a physical point of
view, as showing how energy which is apparently potential may in its
ultimate essence be kinetic.
By means of the formulae (18), which now reduce to
dP[Kappa] dP[Kappa] dP[Kappa]
[chi]` = ---------, [chi]`' = ----------, [chi]`" = ---------- ..., (25)
dP[kappa] dP[kappa]' dP[kappa]"
[Kappa] may also be expressed as a homogeneous quadratic function of
the cyclic velocities [.[chi]], [.[chi]]', [.[chi]]", ... Denoting it
in this form by [Tau]0, we have
[delta]([Tau]0 + [Kappa] = 2[delta][Kappa] = [delta]([kappa] [chi]` + [kappa]'[chi]`' + [kappa]"[chi]`" + ...). (26)
Performing the variations, and omitting the terms which cancel by (2)
and (25), we find
dP[Tau]0 dP[Kappa] dP[Tau]0 dP[Kappa]
-------- = - ---------, -------- = - ---------, ..., (27)
dPq1 dPq1 dPq2 dPq2
so that the formulae (23) become
dP[Tau]0 dP[Tau]0
Q1 = - --------, Q2 = - --------, ... (28)
dPq1 dPq2
A simple example is furnished by the top (MECHANICS, Sec. 22). The cyclic
co-ordinates being [psi], [phi], we find
( mu - [nu] cos [theta]) squared [nu] squared
2[@] = A[theta]` squared, 2[Kappa] = ----------------------- + -----,
A sin squared [theta] C
2[Tau]0 = A sin squared[theta][psi]` squared + C([phi]` + [psi] cos [theta]) squared, (29)
whence we may verify that dP[Tau]0/dP[theta] = - dP[Kappa]/dP[theta]
in accordance with (27). And the condition of equilibrium
dP[Kappa] dPV
--------- = - --------- (30)
dP[theta] dP[theta]
gives the condition of steady precession.
6. _Stability of Steady Motion._
The small oscillations of a conservative system about a configuration
of equilibrium, and the criterion of stability, are discussed in
MECHANICS, Sec. 23. The question of the stability of given types of
motion is more difficult, owing to the want of a sufficiently general,
and at the same time precise, definition of what we mean by
"stability." A number of definitions which have been propounded by
different writers are examined by F. Klein and A. Sommerfeld in their
work _Ueber die Theorie des Kreisels_ (1897-1903). Rejecting previous
definitions, they base their criterion of stability on the character
of the changes produced in the _path_ of the system by small arbitrary
disturbing impulses. If the undisturbed path be the _limiting form_ of
the disturbed path when the impulses are indefinitely diminished, it
is said to be stable, but not otherwise. For instance, the vertical
fall of a particle under gravity is reckoned as stable, although for a
_given_ impulsive disturbance, however small, the deviation of the
particle's position at any time t from the position which it would
have occupied in the original motion increases indefinitely with t.
Even this criterion, as the writers quoted themselves recognize, is
not free from ambiguity unless the phrase "limiting form," as applied
to a path, be strictly defined. It appears, moreover, that a
definition which is analytically precise may not in all cases be easy
to reconcile with geometrical prepossessions. Thus a particle moving
in a circle about a centre of force varying inversely as the cube of
the distance will if slightly disturbed either fall into the centre,
or recede to infinity, after describing in either case a spiral with
an infinite number of convolutions. Each of these spirals has,
analytically, the circle as its limiting form, although the motion in
the circle is most naturally described as unstable.
A special form of the problem, of great interest, presents itself in
the steady motion of a gyrostatic system, when the non-eliminated
co-ordinates q1, q2, ... q_m all vanish (see Sec. 5). This has been
discussed by Routh, Lord Kelvin and Tait, and Poincare. These writers
treat the question, by an extension of Lagrange's method, as a problem
of small oscillations. Whether we adopt the notion of stability which
this implies, or take up the position of Klein and Sommerfeld, there
is no difficulty in showing that stability is ensured if V + [Kappa]
be a minimum as regards variations of q1, q2, ... q_m. The proof is
the same as that of Dirichlet for the case of statical stability.
We can illustrate this condition from the case of the top, where, in
our previous notation,
( mu - [nu]cos [theta]) squared [nu] squared
V + [Kappa] = Mgh cos[theta] + ---------------------- + -----. (1)
2A sin squared [theta] 2C
To examine whether the steady motion with the centre of gravity
vertically above the pivot is stable, we must put mu = [nu]. We then
find without difficulty that V + [Kappa] is a minimum provided [nu] squared
[>=] 4AMgh. The method of small oscillations gave us the condition
[nu] squared > 4AMgh, and indicated instability in the cases [nu] squared [=<]
4AMgh. The present criterion can also be applied to show that the
steady precessional motions in which the axis has a constant
inclination to the vertical are stable.
The question remains, as before, whether it is _essential_ for
stability that V + [Kappa] should be a minimum. It appears that from
the point of view of the theory of small oscillations it is not
essential, and that there may even be stability when V + [Kappa] is a
maximum. The precise conditions, which are of a somewhat elaborate
character, have been formulated by Routh. An important distinction
has, however, been established by Thomson and Tait, and by Poincare,
between what we may call _ordinary_ or _temporary_ stability (which is
stability in the above sense) and _permanent_ or _secular_ stability,
which means stability when regard is had to possible dissipative
forces called into play whenever the co-ordinates q1, q2, ... q_m
vary. Since the total energy of the system at any instant is given (in
the notation of Sec. 5) by an expression of the form [@] + V +
[Kappa], where [@] cannot be negative, the argument of Thomson
and Tait, given under MECHANICS, Sec. 23, for the statical question,
shows that it is a necessary as well as a sufficient condition for
secular stability that V + [Kappa] should be a minimum. When a system
is "ordinarily" stable, but "secularly" unstable, the operation of the
frictional forces is to induce a gradual increase in the amplitude of
the free vibrations which are called into play by accidental
disturbances.
There is a similar theory in relation to the constrained systems
considered in Sec. 3 above. The equation (21) there given leads to the
conclusion that for secular stability of any type of motion in which
the velocities q`1, q`2, ... q`_n are zero it is necessary and
sufficient that the function V - [Tau]0 should be a minimum.
The simplest possible example of this is the case of a particle at the
lowest point of a smooth spherical bowl which rotates with constant
angular velocity ([omega]) about the vertical diameter. This position
obviously possesses "ordinary" stability. If a be the radius of the
bowl, and [theta] denote angular distance from the lowest point, we
have
V - [Tau]0 = mga(1 - cos [theta]) - 1/2m[omega] squareda squared sin squared [theta]; (2)
this is a minimum for [theta] = 0 only so long as [omega] squared < g/a. For
greater values of [omega] the only position of "permanent" stability
is that in which the particle rotates with the bowl at an angular
distance cos^(-1) (g/[omega] squareda) from the lowest point. To examine the
motion in the neighbourhood of the lowest point, when frictional
forces are taken into account, we may take fixed ones, in a horizontal
plane, through the lowest point. Assuming that the friction varies as
the relative velocity, we have
x" = -p squaredx - k(x` + [omega]y), \ (3)
y" = -p squaredy - k(y` - [omega]x), /
where p squared = g/a. These combine into
z" + kz` + (p squared - ik[omega])z = 0, (4)
where z = x + iy, i = [root]-1. Assuming z = Ce^([lambda]t), we find
[lambda] = -1/2k(1 [-+] [omega]/p) +- ip, (5)
if the square of k be neglected. The complete solution is then
x + iy = C1e^(-ss1t)e^(ipt) + C2e^(-ss2t)e^(-ipt), (6)
where ss1 = 1/2k(1 - [omega]/p), ss2 = 1/2k(1 + [omega]/p). (7)
This represents two superposed circular vibrations, in opposite
directions, of period 2[pi]/p. If [omega] < p, the amplitude of each
of these diminishes asymptotically to zero, and the position x = 0, y
= 0 is permanently stable. But if [omega] > p the amplitude of that
circular vibration which agrees in sense with the rotation [omega]
will continually increase, and the particle will work its way in an
ever-widening spiral path towards the eccentric position of secular
stability. If the bowl be not spherical but ellipsoidal, the vertical
diameter being a principal axis, it may easily be shown that the
lowest position is permanently stable only so long as the period of
the rotation is longer than that of the slower of the two normal
modes in the absence of rotation (see MECHANICS, Sec. 13).
7. _Principle of Least Action._
Stationary Action.
The preceding theories give us statements applicable to the system at
any one instant of its motion. We now come to a series of theorems
relating to the whole motion of the system between any two
configurations through which it passes, viz. we consider the actual
motion and compare it with other imaginable motions, differing
infinitely little from it, between the same two configurations. We use
the symbol [delta] to denote the transition from the actual to any one
of the hypothetical motions.
The best-known theorem of this class is that of _Least Action_,
originated by P.L.M. de Maupertuis, but first put in a definite form
by Lagrange. The "action" of a single particle in passing from one
position to another is the space-integral of the momentum, or the
time-integral of the _vis viva_. The action of a dynamical system is
the sum of the actions of its constituent particles, and is
accordingly given by the formula
_ _ _
/ / /
A = [Sigma] | mvds = [Sigma] | mv squareddt = 2 | [Tau]dt. (1)
_/ _/ _/
The theorem referred to asserts that the free motion of a conservative
system between any two given configurations is characterized by the
property
[delta]A = 0, (2)
provided the total energy have the same constant value in the varied
motion as in the actual motion.
If t, t' be the times of passing through the initial and final
configurations respectively, we have
_
/ t'
[delta]A = [delta] | [Sigma]m(x` squared + y` squared + z` squared)dt
_/t
_
/ t'
= 2 | [delta][Tau]dt + 2[Tau]'[delta]t' + 2[Tau][delta]t, (3)
_/t
since the upper and lower limits of the integral must both be regarded
as variable. This may be written
_ _
/ t' / t'
[delta]A = | [delta][Tau]dt + | [Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t
_/t _/t
_ _ _
/ t' | | t'
= | [delta][Tau]dt + | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z |
_/t |_ _| t
_
/ t'
- | [Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z)dt + 2[Tau]'[delta]t' - 2[Tau][delta]t. (4)
_/t
Now, by d'Alembert's principle,
[Sigma]m( x"[delta]x + y"[delta]y + z"[delta]z ) = -[delta]V, (5)
and by hypothesis we have
[delta]([Tau] + V) = 0. (6)
The formula therefore reduces to
_ _
| |t'
[delta]A = | [Sigma]m (x`[delta]x + y`[delta]y + z`[delta]z) | + 2[Tau]'[delta]t' - 2[Tau][delta]t. (7)
|_ _|t
Since the terminal configurations are unaltered, we must have at the
lower limit
[delta]x + x`[delta]t = 0, [delta]y + y`[delta]t = 0, [delta]z + z`[delta]t = 0, (8)
with similar relations at the upper limit. These reduce (7) to the
form (2).
The equation (2), it is to be noticed, merely expresses that the
variation of A vanishes _to the first order_; the phrase _stationary
action_ has therefore been suggested as indicating more accurately
what has been proved. The action in the free path between two given
configurations is in fact not invariably a minimum, and even when a
minimum it need not be the _least possible_ subject to the given
conditions. Simple illustrations are furnished by the case of a single
particle. A particle moving on a smooth surface, and free from
extraneous force, will have its velocity constant; hence the theorem
in this case resolves itself into
_
/
[delta] | ds = 0, (9)
_/
i.e. the path must be a geodesic line. Now a geodesic is not
necessarily the _shortest_ path between two given points on it; for
example, on the sphere a great-circle arc ceases to be the shortest
path between its extremities when it exceeds 180 deg.. More generally,
taking any surface, let a point P, starting from O, move along a
geodesic; this geodesic will be a minimum path from O to P until P
passes through a point O' (if such exist), which is the intersection
with a consecutive geodesic through O. After this point the minimum
property ceases. On an anticlastic surface two geodesics cannot
intersect more than once, and each geodesic is therefore a minimum
path between any two of its points. These illustrations are due to
K.G.J. Jacobi, who has also formulated the general criterion,
applicable to all dynamical systems, as follows:--Let O and P denote
any two configurations on a natural path of the system. If this be the
sole free path from O to P with the prescribed amount of energy, the
action from O to P is a minimum. But if there be several distinct
paths, let P vary from coincidence with O along the first-named path;
the action will then cease to be a minimum when a configuration O' is
reached such that two of the possible paths from O to O' coincide. For
instance, if O and P be positions on the parabolic path of a
projectile under gravity, there will be a second path (with the same
energy and therefore the same velocity of projection from O), these
two paths coinciding when P is at the other extremity (O', say) of the
focal chord through O. The action from O to P will therefore be a
minimum for all positions of P short of O'. Two configurations such as
O and O' in the general statement are called conjugate _kinetic foci_.
Cf. VARIATIONS, CALCULUS OF.
Before leaving this topic the connexion of the principle of stationary
action with a well-known theorem of optics may be noticed. For the
motion of a particle in a conservative field of force the principle
takes the form
_
/
[delta] | vds = 0. (10)
_/
On the corpuscular theory of light v is proportional to the refractive
index mu of the medium, whence
_
/
[delta] | muds = 0. (11)
_/
Hamiltonian principle.
In the formula (2) the energy in the hypothetical motion is
prescribed, whilst the time of transit from the initial to the final
configuration is variable. In another and generally more convenient
theorem, due to Hamilton, the time of transit is prescribed to be the
same as in the actual motion, whilst the energy may be different and
need not (indeed) be constant. Under these conditions we have
_
/t'
[delta] | ([Tau] - V)dt = 0, (12)
_/t
where t, t' are the prescribed times of passing through the given
initial and final configurations. The proof of (12) is simple; we have
_ _ _
/t' /t' /t'
[delta] | ([Tau] - V)dt = | ([delta][Tau] - [delta]V)dt = | {[Sigma]m(x`[delta]x` + y`[delta]y` + z`[delta]z`) - [delta]V}dt
_/t _/t _/t
_ _
| |t'
= | [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) |
|_ _|t
_
/t'
- | {[Sigma]m(x"[delta]x + y"[delta]y + z"[delta]z) + [delta]V}dt (13)
_/t
The integrated terms vanish at both limits, since by hypothesis the
configurations at these instants are fixed; and the terms under the
integral sign vanish by d'Alembert's principle.
The fact that in (12) the variation does not affect the time of
transit renders the formula easy of application in any system of
co-ordinates. Thus, to deduce Lagrange's equations, we have
_ _
/t' /t'/dP[Tau] dP[Tau] dPV \
| ([delta][Tau]-[delta]V)dt = | ( -------[delta]q`1 + -------[delta]q1 + ... - ----[delta]q1 - ...)dt
_/t _/t \ dPq`1 dPq1 dPq1 /
_ _
| |t'
= | p1[delta]q1 + p2[delta]q2 + ... |
|_ _|t
_ _ _
/t'| / dP[Tau] dPV \ / dP[Tau] dPV\ |
- | | (p`1 - ------- + ---- )[delta]q1 + (p`2 - ------- + -----)[delta]q2 + ...|dt. (14)
_/t |_ \ dPq1 dPq1/ \ dPq2 dPq2/ _|
The integrated terms vanish at both limits; and in order that the
remainder of the right-hand member may vanish it is necessary that the
coefficients of [delta]q1, [delta]q2, ... under the integral sign
should vanish for all values of t, since the variations in question
are independent, and subject only to the condition of vanishing at the
limits of integration. We are thus led to Lagrange's equation of
motion for a conservative system. It appears that the formula (12) is
a convenient as well as a compact embodiment of the whole of ordinary
dynamics.
Extension to cyclic systems.
The modification of the Hamiltonian principle appropriate to the case
of cyclic systems has been given by J. Larmor. If we write, as in Sec. 1
(25),
R = T - [kappa][chi]` - [kappa]'[chi]`' - [kappa]''[chi]`" - ..., (15)
we shall have
_
/t'
[delta] | (R - V)dt = 0, (16)
_/t
provided that the variation does not affect the cyclic momenta
[kappa], [kappa]', [kappa]", ..., and that the configurations at times
t and t' are unaltered, so far as they depend on the palpable
co-ordinates q1, q2, ... q_m. The initial and final values of the
ignored co-ordinates will in general be affected.
To prove (16) we have, on the above understandings,
_ _
/t' /t'
[delta] | (R - V)dt = | ([delta][Tau] - [kappa][delta][chi]` - ... -[delta]V)dt
_/t _/t
_
/t' /dP[Tau] dP[Tau] \
= | ( -------[delta]q`1 + ... + -------[delta]q1 + ... - [delta]V )dt, (17)
_/t \ dPq`1 dPq1 /
where terms have been cancelled in virtue of Sec. 5 (2). The last member
of (17) represents a variation of the integral
_
/t'
| ([Tau] - V)dt
_/t
on the supposition that [delta]X = 0, [delta]X' = 0, [delta]X" = 0,
... throughout, whilst [delta]q1, [delta]q2, [delta]q_m vanish at
times t and t'; i.e. it is a variation in which the initial and final
configurations are absolutely unaltered. It therefore vanishes as a
consequence of the Hamiltonian principle in its original form.
Larmor has also given the corresponding form of the principle of least
action. He shows that if we write
_
/
A = |(2[Tau] - [kappa][chi]` - [kappa]'[chi]`' - [kappa]"[chi]`" - ...)dt, (18)
_/
then
[delta]A = 0, (19)
provided the varied motion takes place with the same constant value of
the energy, and with the same constant cyclic momenta, between the
same two configurations, these being regarded as defined by the
palpable co-ordinates alone.
Sec. 8. _Hamilton's Principal and Characteristic Functions._
Principal function.
In the investigations next to be described a more extended meaning is
given to the symbol [delta]. We will, in the first instance, denote by
it an infinitesimal variation of the most general kind, affecting not
merely the values of the co-ordinates at any instant, but also the
initial and final configurations and the times of passing through
them. If we put
_
/t'
S = | (T - V)dt, (1)
_/t
we have, then,
_
/t'
[delta]S = (T' - V')[delta]t' - (T - V)[delta]t + | ([delta]T - [delta]V)dt
_/t
_ _
| |t'
= (T' - V')[delta]t' - (T - V)[delta]t + |[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z)| (2)
|_ _|t
Let us now denote by x' + [delta]x', y' + [delta]y', z' + [delta]z',
the final co-ordinates (i.e. at time t' + [delta]t') of a particle m.
In the terms in (2) which relate to the upper limit we must therefore
write [delta]x' - x`'[delta]t', [delta]y' - y`'[delta]t',
[delta]z' - z`'[delta]t' for [delta]x, [delta]y, [delta]z. With a
similar modification at the lower limit, we obtain
[delta]S = - H[delta][tau] + [Sigma]m(x`'[delta]x' + y`'[delta]y' + z`'[delta]z')
- [Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z), (3)
where H(= T + V) is the constant value of the energy in the free
motion of the system, and [tau](= t' - t) is the time of transit. In
generalized co-ordinates this takes the form
[delta]S = - H[delta][tau] + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (4)
Now if we select any two arbitrary configurations as initial and
final, it is evident that we can in general (by suitable initial
velocities or impulses) start the system so that it will of itself
pass from the first to the second in any prescribed time [tau]. On
this view of the matter, S will be a function of the initial and final
co-ordinates (q1, q2, ... and q'1, q'2, ...) and the time [tau], as
independent variables. And we obtain at once from (4)
dPS dPS \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (5)
dPS dPS |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /
dPS
and H = - -------. (6)
dP[tau]
S is called by Hamilton the _principal function_; if its general form
for any system can be found, the preceding equations suffice to
determine the motion resulting from any given conditions. If we
substitute the values of p1, p2, ... and H from (5) and (6) in the
expression for the kinetic energy in the form [Tau]' (see Sec. 1), the
equation
Tš + V = H (7)
becomes a partial differential equation to be satisfied by S. It has
been shown by Jacobi that the dynamical problem resolves itself into
obtaining a "complete" solution of this equation, involving n + 1
arbitrary constants. This aspect of the subject, as a problem in
partial differential equations, has received great attention at the
hands of mathematicians, but must be passed over here.
Characteristic function.
There is a similar theory for the function
_
/
A = 2 | Tdt = S + H[tau] (8)
_/
It follows from (4) that
[delta]A = [tau][delta]H + p'1[delta]q'1 + p'2[delta]q'2 + ...
- p1[delta]q1 - p2[delta]q2 - .... (9)
This formula (it may be remarked) contains the principle of "least
action" as a particular case. Selecting, as before, any two arbitrary
configurations, it is in general possible to start the system from one
of these, with a prescribed value of the total energy H, so that it
shall pass through the other. Hence, regarding A as a function of the
initial and final co-ordinates and the energy, we find
dPA dPA \
p'1 = -----, p'2 = -----, ..., |
dPq'1 dPq'2 |
> (10)
dPA dPA |
p1 = - ----, p2 = - ----, ..., |
dPq1 dPq2 /
dPA
and [tau] = --- (11)
dPH
A is called by Hamilton the _characteristic function_; it represents,
of course, the "action" of the system in the free motion (with
prescribed energy) between the two configurations. Like S, it
satisfies a partial differential equation, obtained by substitution
from (10) in (7).
The preceding theorems are easily adapted to the case of cyclic
systems. We have only to write
_ _
/t' /t'
S = | (R - V)dt= | (T - [kappa][chi]` - [kappa]'[chi]`' - ... - V)dt (12)
_/t _/t
in place of (1), and
_
/
A = | (2T - [kappa][chi]` - [kappa]'[chi]`' - ...)dt, (3)
_/
in place of (8); cf. Sec. 7 ad fin. It is understood, of course, that in
(12) S is regarded as a function of the initial and final values of
the palpable co-ordinates q1, q2, ... q_m, and of the time of transit
[tau], the cyclic momenta being invariable. Similarly in (13), A is
regarded as a function of the initial and final values of q1, q2, ...
q_m, and of the total energy H, with the cyclic momenta invariable. It
will be found that the forms of (4) and (9) will be conserved,
provided the variations [delta]q1, [delta]q2, ... be understood to
refer to the palpable co-ordinates alone. It follows that the
equations (5), (6) and (10), (11) will still hold under the new
meanings of the symbols.
9. _Reciprocal Properties of Direct and Reversed Motions._
Lagrange's formula.
We may employ Hamilton's principal function to prove a very remarkable
formula connecting any _two_ slightly disturbed natural motions of the
system. If we use the symbols [delta] and [Delta] to denote the
corresponding variations, the theorem is
d
--[Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r) = 0; (1)
dt
or integrating from t to t',
[Sigma]([delta]p'_r.[Delta]q'_r - [Delta]q'_r.[delta]q'_r) = [Sigma]([delta]p_r.[Delta]q_r - [Delta]p_r.[delta]q_r). (2)
If for shortness we write
dP squaredS dP squaredS
(r,s) = ----------, (r,s') = -----------, (3)
dPq_rdPq_s dPq_rdPq'_s
we have
dPp_r = - [Sigma]_s(r,s)[delta]q_s - [Sigma]_s(r,s')[delta]q'_s (4)
with a similar expression for [Delta]p_r. Hence the right-hand side of
(2) becomes
- [Sigma]_r{[Sigma]_s(r,s)[delta]q_s + [Sigma]_s(r,s')[delta]q'_s}[Delta]q_r
+ [Sigma]_r{[Sigma]_s(r,s)[Delta]q_s + [Sigma]_s(r,s')[Delta]q'_s}[delta]q_r
= [Sigma]_r[Sigma]_s(r,s'){[delta]q_r.[Delta]q'_s - [Delta]q_r.[delta]q'_s}. (5)
The same value is obtained in like manner for the expression on the
left hand of (2); hence the theorem, which, in the form (1), is due to
Lagrange, and was employed by him as the basis of his method of
treating the dynamical theory of _Variation of Arbitrary Constants_.
Helmholtz's reciprocal theorems.
The formula (2) leads at once to some remarkable reciprocal relations
which were first expressed, in their complete form, by Helmholtz.
Consider any natural motion of a conservative system between two
configurations O and O' through which it passes at times t and t'
respectively, and let t' - t = [tau]. As the system is passing through
O let a small impulse [delta]p_r be given to it, and let the
consequent alteration in the co-ordinate q_s after the time [tau] be
[delta]q'_s. Next consider the _reversed_ motion of the system, in
which it would, if undisturbed, pass from O' to O in the same time
[tau]. Let a small impulse [delta]p'_s be applied as the system is
passing through O', and let the consequent change in the co-ordinate
q_r after a time [tau] be [delta]q_r. Helmholtz's first theorem is to
the effect that
[delta]q_r : [delta]p'_s = [delta]q'_s : [delta]p_r. (6)
To prove this, suppose, in (2), that all the [delta]q vanish, and
likewise all the [delta]p with the exception of [delta]p_r. Further,
suppose all the [Delta]q' to vanish, and likewise all the [Delta]p'
except [Delta]p'_s, the formula then gives
[delta]p_r.[Delta]q_r = - [Delta]p'_s.[delta]q'_s, (7)
which is equivalent to Helmholtz's result, since we may suppose the
symbol [Delta] to refer to the reversed motion, provided we change
the signs of the [Delta]p. In the most general motion of a top
(MECHANICS, Sec. 22), suppose that a small impulsive couple about the
vertical produces after a time [tau] a change [delta][theta] in the
inclination of the axis, the theorem asserts that in the reversed
motion an equal impulsive couple in the plane of [theta] will produce
after a time [tau] a change [delta][psi], in the azimuth of the axis,
which is equal to [delta][theta]. It is understood, of course, that
the couples have no components (in the generalized sense) except of
the types indicated; for instance, they may consist in each case of a
force applied to the top at a point of the axis, and of the
accompanying reaction at the pivot. Again, in the corpuscular theory
of light let O, O' be any two points on the axis of a symmetrical
optical combination, and let V, V' be the corresponding velocities of
light. At O let a small impulse be applied perpendicular to the axis
so as to produce an angular deflection [delta][theta], and let ss'
be the corresponding lateral deviation at O'. In like manner in the
reversed motion, let a small deflection [delta][theta]' at O' produce
a lateral deviation ss at O. The theorem (6) asserts that
ss ss'
----------------- = ---------------, (8)
V'[delta][theta]' V[delta][theta]
or, in optical language, the "apparent distance" of O from O' is to
that of O' from O in the ratio of the refractive indices at O' and O
respectively.
Helmholtz's second reciprocal theorem.
In the second reciprocal theorem of Helmholtz the configuration O is
slightly varied by a change [delta]q_r in one of the co-ordinates, the
momenta being all unaltered, and [delta]q'_s is the consequent
variation in one of the momenta after time [tau]. Similarly in the
reversed motion a change [delta]p'_s produces after time [tau] a
change of momentum [delta]p_r. The theorem asserts that
[delta]p'_s : [delta]q_r = [delta]p_r : [delta]q'_s (9)
This follows at once from (2) if we imagine all the [delta]p to
vanish, and likewise all the [delta]q save [delta]q_r, and if
(further) we imagine all the [Delta]p' to vanish, and all the
[Delta]q' save [Delta]q'_s. Reverting to the optical illustration, if
F, F', be principal foci, we can infer that the convergence at F' of a
parallel beam from F is to the convergence at F of a parallel beam
from F' in the inverse ratio of the refractive indices at F' and F.
This is equivalent to Gauss's relation between the two principal focal
lengths of an optical instrument. It may be obtained otherwise as a
particular case of (8).
We have by no means exhausted the inferences to be drawn from
Lagrange's formula. It may be noted that (6) includes as particular
cases various important reciprocal relations in optics and acoustics
formulated by R.J.E. Clausius, Helmholtz, Thomson (Lord Kelvin) and
Tait, and Lord Rayleigh. In applying the theorem care must be taken
that in the reversed motion the reversal is complete, and extends to
every velocity in the system; in particular, in a cyclic system the
cyclic motions must be imagined to be reversed with the rest.
Conspicuous instances of the failure of the theorem through incomplete
reversal are afforded by the propagation of sound in a wind and the
propagation of light in a magnetic medium.
It may be worth while to point out, however, that there is no such
limitation to the use of Lagrange's formula (1). In applying it to
cyclic systems, it is convenient to introduce conditions already laid
down, viz. that the co-ordinates q_r are the palpable co-ordinates and
that the cyclic momenta are invariable. Special inference can then be
drawn as before, but the interpretation cannot be expressed so neatly
owing to the non-reversibility of the motion.
AUTHORITIES.--The most important and most accessible early authorities
are J.L. Lagrange, _Mecanique analytique_ (1st ed. Paris, 1788, 2nd
ed. Paris, 1811; reprinted in _Oeuvres_, vols. xi., xii., Paris,
1888-89); Hamilton, "On a General Method in Dynamics," _Phil. Trans._
1834 and 1835; C.G.J. Jacobi, _Vorlesungen ueber Dynamik_ (Berlin,
1866, reprinted in _Werke_, Supp.-Bd., Berlin, 1884). An account of
the extensive literature on the differential equations of dynamics and
on the theory of variation of parameters is given by A. Cayley,
"Report on Theoretical Dynamics," _Brit. Assn. Rep._ (1857),
_Mathematical Papers_, vol. iii. (Cambridge, 1890). For the modern
developments reference may be made to Thomson and Tait, _Natural
Philosophy_ (1st ed. Oxford, 1867, 2nd ed. Cambridge, 1879); Lord
Rayleigh, _Theory of Sound_, vol. i. (1st ed. London, 1877; 2nd ed.
London, 1894); E.J. Routh, _Stability of Motion_ (London, 1877), and
_Rigid Dynamics_ (4th ed. London, 1884); H. Helmholtz, "Ueber die
physikalische Bedeutung des Prinzips der kleinsten Action," _Crelle_,
vol. c., 1886, reprinted (with other cognate papers) in _Wiss. Abh._
vol. iii. (Leipzig, 1895); J. Larmor, "On Least Action," _Proc. Lond.
Math. Soc._ vol. xv. (1884); E.T. Whittaker, _Analytical Dynamics_
(Cambridge, 1904). As to the question of stability, reference may be
made to H. Poincare, "Sur l'equilibre d'une masse fluide animee d'un
mouvement de rotation" _Acta math._ vol. vii. (1885); F. Klein and A.
Sommerfeld, _Theorie des Kreisels_, pts. 1, 2 (Leipzig, 1897-1898); A.
Lioupanoff and J. Hadamard, _Liouville_, 5me serie, vol. iii. (1897);
T.J.I. Bromwich, Proc. Lond. Math. Soc. vol. xxxiii. (1901). A
remarkable interpretation of various dynamical principles is given by
H. Hertz in his posthumous work _Die Prinzipien der Mechanik_
(Leipzig, 1894), of which an English translation appeared in 1900.
(H. Lb.)
DYNAMITE (Gr. [Greek: dynamis], power), the name given to several
explosive preparations containing nitroglycerin (q.v.) which are almost
exclusively used for blasting purposes. The first practical application
of nitroglycerin in this way was made by A. Nobel in 1863. He soaked
gunpowder with the liquid and fired the gunpowder by an ordinary fuse.
Later he found that nitroglycerin could be detonated by the explosion of
several materials such as fulminate of mercury, the use of which as a
detonator he patented in 1867. In 1866-1867 he experimented with
charcoal and other substances, and found the infusorial earth known as
kieselguhr, which consists mainly of silica (nearly 95%), eminently
adapted to the purpose, as it was inert, non-combustible, and after a
little heating and preparation very porous, retaining a large amount of
nitroglycerin as water is held in a sponge, without very serious
exudation on standing. This kieselguhr dynamite is generally made by
incorporating three parts of nitroglycerin with one part of the dry
earth, the paste being then formed into cylindrical cartridges. This
work is done by hand. Generally a small percentage of the kieselguhr is
replaced by a mixture containing sodium and ammonium carbonates, talc
and ochre. This product is known as dynamite No. 1. Disabilities
attaching to kieselguhr dynamite are that when placed in water the
nitroglycerin is liable to be exuded or displaced, also that, like
nitroglycerin itself, it freezes fairly easily and thawing the frozen
cartridges is a dangerous operation. Other substances, e.g. kaolin,
tripoli, magnesia alba (magnesium carbonate), alumina, sugar, charcoal,
some powdered salts and mixtures of sawdust and salts, have been shown
to be absorbents more or less adapted to the purpose of making a
dynamite. Charcoal from cork is said to absorb about 90% of its weight
of nitroglycerin. With the idea of obtaining greater safety, mixtures
have been made of nitroglycerin with wood fibre, charcoal and metallic
nitrates. Lithofracteur, for instance, consists of 50% nitroglycerin and
a mixture of prepared sawdust, kieselguhr and barium nitrate. Carbonite
contains 25% of nitroglycerin, the remainder being a mixture of
wood-meal and alkali nitrates, with about 1% of sulphur. Dualin, atlas
dynamite and potentite are other modifications.
A convenient form in which nitroglycerin can be made up for blasting
purposes, especially in wet ground, is the gelatinous material obtained
by the action of nitroglycerin, either alone or with the help of
solvents, on low-grade or soluble gun-cottons. It is known as blasting
gelatin, and was first made by Nobel by incorporating 6 or 7% of low
nitrated cellulose (collodion cotton or soluble gun-cotton) with
slightly warmed nitroglycerin. The result is a transparent plastic
material, of specific gravity 1.5 to 1.6, which may be kept under water
for a long time without appreciable change. It is less sensitive to
detonation than ordinary dynamite, and although its explosion is
slightly slower it is more powerful than dynamite and much superior to
the liquid nitroglycerin. Blasting gelatin also freezes and is sensitive
to percussion in this state. Camphor and other substances have been
added to blasting gelatin to render it more solid and less sensitive.
Some modifications of blasting gelatin, e.g. gelignite, contain
wood-meal and such oxygen-containing salts as potassium nitrate.
Experience has conclusively shown that dynamites are more satisfactory,
quicker, and more intense in action than liquid nitroglycerin.
To prevent nitroglycerin and some of the forms of dynamite from freezing
it has been proposed to add to them small quantities of either
monochlor-dinitroglycerin or of a nitrated poly-glycerin. The former is
obtained by first acting upon glycerin with hydrogen chloride to produce
_u-_chlorhydrin or chlor-propylene glycol, C3H7O2Cl, which is then
nitrated as in the case of glycerin. The latter is obtained by heating
glycerin for six or seven hours to about 300 deg. C., whereby water is split
off in such manner that a diglycerin C6H14O5, for the most part,
results. This on nitration in the usual manner gives a product
C6H{10}N4O{13}, which burns and explodes in a similar manner to ordinary
nitroglycerin, but is less sensitive and does not so easily freeze. The
mono- and di-nitrates of glycerin have also been proposed as additions
to ordinary nitroglycerin (q.v.) for the same purpose. (W. R. E. H.)
DYNAMO (a shortened form of "dynamo-electric machine," from Gr. [Greek:
dynamis], power), a machine for converting mechanical into electrical
energy.
The dynamo ranks with the telegraph and telephone as one of the three
striking applications of electrical and magnetic science to which the
material progress that marked the second half of the 19th century was in
no small measure due. Since the discovery of the principle of the dynamo
by Faraday in 1831 the simple model which he first constructed has been
gradually developed into the machines of 5000 horse-power or more which
are now built to meet the needs of large cities for electric lighting
and power, while at the same time the numbers of dynamos in use have
increased almost beyond estimate. Yet such was the insight of Faraday
into the fundamental nature of the dynamo that the theory of its action
which he laid down has remained essentially unchanged. His experiments
on the current which was set up in a coil of wire during its movement
across the poles of a magnet led naturally to the explanation of induced
electromotive force as caused by the linking or unlinking of magnetic
lines of flux with an electric circuit. For the more definite case of
the dynamo, however, we may, with Faraday, make the transition from
line-linkage to the equivalent conception of "line-cutting" as the
source of E.M.F.--in other words, to the idea of electric conductors
"cutting" or intersecting[1] the lines of flux in virtue of relative
motion of the magnetic field and electric circuit. On the 28th of
October 1831 Faraday mounted a copper disk so that it could be rotated
edgewise between the poles of a permanent horse-shoe magnet. When so
rotated, it cut the lines of flux which passed transversely through its
lower half, and by means of two rubbing contacts, one on its periphery
and the other on its spindle, the circuit was closed through a
galvanometer, which indicated the passage of a continuous current so
long as the disk was rotated (fig. 1). Thus by the invention of the
first dynamo Faraday proved his idea that the E.M.F. induced through the
interaction of a magnetic field and an electric circuit was due to the
passage of a portion of the electric circuit _across_ the lines of flux,
or vice versa, and so could be maintained if the cutting of the lines
were made continuous.[2] In comparison with Faraday's results, the
subsequent advance is to be regarded as a progressive perfecting of the
mechanical and electro-magnetic design, partly from the theoretical and
partly from the practical side, rather than as modifying or adding to
the idea which was originally present in his mind, and of which he
already saw the possibilities.
[Illustration: FIG. 1.]
A dynamo, then, is a machine in which, by means of continuous relative
motion, an electrical conductor or system of conductors forming part of
a circuit is caused to cut the lines of a magnetic field or fields; the
cutting of the magnetic flux induces an electromotive force in the
conductors, and when the circuit is closed a current flows, whereby
mechanical energy is converted into electrical energy.
Little practical use could be made of electrical energy so long as its
only known sources were frictional machines and voltaic batteries. The
cost of the materials for producing electrical currents on a large
scale by chemical action was prohibitive, while the frictional machine
only yielded very small currents at extremely high potentials. In the
dynamo, on the other hand, electrical energy in a convenient form
could be cheaply and easily obtained by mechanical means, and with its
invention the application of electricity to a wide range of commercial
purposes became economically possible. As a converter of energy from
one form to another it is only surpassed in efficiency by another
electrical appliance, namely, the transformer (see TRANSFORMERS). In
this there is merely conversion of electrical energy at a high
potential into electrical energy at a low potential, or vice versa,
but in the dynamo the mechanical energy which must be applied to
maintain the relative movement of magnetic field and conductor is
absorbed, and reappears in an electrical form. A true transformation
takes place, and the proportion which the rate of delivery of
electrical energy bears to the power absorbed, or in other words the
_efficiency_, is the more remarkable. The useful return or "output" at
the terminals of a large machine may amount to as much as 95% of the
mechanical energy which forms the "input." Since it needs some prime
mover to drive it, the dynamo has not made any direct addition to our
sources of energy, and does not therefore rank with the primary
battery or oil-engine, or even the steam-engine, all of which draw
their energy more immediately from nature. Yet by the aid of the
dynamo the power to be derived from waterfalls can be economically and
conveniently converted into an electrical form and brought to the
neighbouring factory or distant town, to be there reconverted by
motors into mechanical power. Over any but very short distances energy
is most easily transmitted when it is in an electrical form, and
turbine-driven dynamos are very largely and successfully employed for
such transmission. Thus by conducing to the utilization of water-power
which may previously have had but little value owing to its
disadvantageous situation, the dynamo may almost be said to have added
another to our available natural resources.
The two essential parts of the dynamo, as required by its definition,
may be illustrated by the original disk machine of Faraday. They are (1)
the _iron magnet_, between the poles of which a magnetic field exists,
and (2) the _electrical conductors_, represented by the rotating copper
disk. The sector of the disk cutting the lines of the field forms part
of a closed electric circuit, and has an E.M.F. induced in it, by reason
of which it is no longer simply a conductor, but has become "active." In
its more highly developed form the simple copper disk is elaborated into
a system of many active wires or bars which form the "winding," and
which are so interconnected as to add up their several E.M.F.'s. Since
these active wires are usually mounted on an iron structure, which may
be likened to the keeper or "armature" of a magnet rotating between its
poles, the term "armature" has been extended to cover not only the iron
core, but also the wires on it, and when there is no iron core it is
even applied to the copper conductors themselves. In the dynamo of
Faraday the "armature" was the rotating portion, and such is the case
with modern continuous-current dynamos; in alternators, however, the
magnet, or a portion of it, is more commonly rotated while the armature
is stationary. It is in fact immaterial to the action whether the one or
the other is moved, or both, so long as their relative motion causes the
armature conductors to cut the magnetic flux. As to the ultimate reason
why an E.M.F. should be thereby induced, physical science cannot as yet
yield any surer knowledge than in the days of Faraday.[3] For the
engineer, it suffices to know that the E.M.F. of the dynamo is due to
the cutting of the magnetic flux by the active wires, and, further, is
proportional to the rate at which the lines are cut.[4]
[Illustration: FIG. 2.]
The equation of the _electromotive force_ which is required in order to
render this statement quantitative must contain three factors, namely,
the density of the flux in the air-gap through which the armature
conductors move, the active length of these wires, and the speed of
their movement. For given values of the first and third factors and a
single straight wire moved parallel to itself through a uniform field,
the maximum rate of cutting is evidently obtained when the three
directions of the lines of the conductor's length and of the relative
motion are respectively at right angles to each other, as shown by the
three co-ordinate axes of fig. 2. The E.M.F. of the single wire is then
E = B_gLV x 10^(-8) volts (1)
where B_g is the density of the flux within the air-gap expressed in
C.G.S. lines per square centimetre, L is the active length of the
conductor within the field in centimetres, and V is the velocity of
movement in centimetres per second. Further, the direction in which the
E.M.F. has the above maximum value is along the length of the conductor,
its "sense" being determined by the direction of the movement[5] in
relation to the direction of the field.
The second fundamental equation of the dynamo brings to light its
mechanical side, and rests on H.C. Oersted's discovery of the
interaction of a magnetic field and an electric current. If a straight
electric conductor through which a current is passing be so placed in a
magnetic field that its length is not parallel to the direction of the
lines of flux, it is acted on by a force which will move it, if free, in
a definite direction relatively to the magnet; or if the conductor is
fixed and the magnet is free, the latter will itself move in the
opposite direction. Now in the dynamo the active wires are placed so
that their length is at right angles to the field; hence when they are
rotated and an electric current begins to flow under the E.M.F. which
they induce, a mutual force at once arises between the copper conductors
and the magnet, and the direction of this force must by Lenz's law be
opposed to the direction of the movement. Thus as soon as the disk of
fig. 1 is rotated and its circuit is closed, it experiences a mechanical
pull or drag which must be overcome by the force applied to turn the
disk. While the magnet must be firmly held so as to remain stationary,
the armature must be of such mechanical construction that its wires can
be forcibly driven through the magnetic field against the mutual pull.
This law of electrodynamic action may be quantitatively stated in an
_equation of mechanical force_, analogous to the equation (I.) of
electromotive force, which states the law of electromagnetic induction.
If a conductor of length L cm., carrying a current C amperes, is
immersed in a field of uniform density B_g, and the length of the
conductor is at right angles to the direction of the lines, it is acted
on by a force
F = B_gLC x 10^(-1) dynes, (2)
and the direction of this force is at right angles to the conductor and
to the field. The rate at which electrical energy is developed, when
this force is overcome by moving the conductor as a dynamo through the
field, is EC = B_gLVC x 10^(-8) watts, whence the equality of the
mechanical power absorbed and the electrical power developed (as
required by the law of the conservation of energy) is easily
established. The whole of this power is not, however, available at the
terminals of the machine; if R_a be the resistance of the armature in
ohms, the passage of the current C_a through the armature conductors
causes a drop of pressure of C_aR_a volts, and a corresponding loss of
energy in the armature at the rate of C_a squaredR_a watts. As the resistance
of the external circuit R_e is lowered, the current C = E_a/(R_e + R_a)
is increased. The increase of the current is, however, accompanied by a
progressive increase in the loss of energy over the armature, and as
this is expended in heating the armature conductors, their temperature
may rise so much as to destroy the insulating materials with which they
are covered. Hence the temperature which the machine may be permitted to
attain in its working is of great importance in determining its output,
the current which forms one factor therein being primarily limited by
the heating which it produces in the armature winding. The lower the
resistance of the armature, the less the rise of its temperature for a
given current flowing through it; and the reason for the almost
universal adoption of copper as the material for the armature conductors
is now seen to lie in its high conductivity.[6]
Since the voltage of the dynamo is the second factor to which its output
is proportional, the conditions which render the induced E.M.F. a
maximum must evidently be reproduced as far as possible in practice, if
the best use is to be made of a given mass of iron and copper. The first
problem, therefore, in the construction of the dynamo is the disposition
of the wires and field in such a manner that the three directions of
field, length of active conductors, and movement are at right angles to
one another, and so that the relative motion is continuous.
Reciprocating motion, such as would be obtained by direct attachment of
the conductors to the piston of a steam-engine, has been successfully
employed only in the special case of an "oscillator,"[7] producing a
small current very rapidly changing in direction. Rotary motion is
therefore universally adopted, and with this two distinct cases arise.
Either (A) the active length of the wire is parallel to the axis of
rotation, or (B) it is at right angles to it.
[Illustration: FIG. 3.]
(A) If a conductor is rotated in the gap between the poles of a
horse-shoe magnet, and these poles have plane parallel faces opposing
one another as in fig. 3, not only is the density of the flux in the
interpolar gap small, but the direction of movement is not always at
right angles to the direction of the lines, which for the most part pass
straight across from one opposing face to the other. When the conductor
is midway between the poles (i.e. either at its highest or lowest
point), it is at this instant sliding along the lines and does not cut
them, so that its E.M.F. is zero. Taking this position as the
starting-point, as the conductor moves round, its rate of line-cutting
increases to a maximum when it has moved through a right angle and is
opposite to the centre of a pole-face (as in fig. 3), from which point
onward the rate decreases to zero when it has moved through 180 deg.. Each
time the conductor crosses a line drawn symmetrically through the gap
between the poles and at right angles to the axis of rotation, the
E.M.F. along its length is reversed in direction, since the motion
relatively to the direction of the field is reversed. If the ends of the
active conductor are electrically connected to two collecting rings
fixed upon, but insulated from, the shaft, two stationary brushes bb can
be pressed on the rings so as to make a sliding contact. An external
circuit can then be connected to the brushes, which will form the
"terminals" of the machine, the periodically reversed or alternating
E.M.F. induced in the active conductor will cause an alternating current
to flow through conductor and external circuit, and the simplest form of
"alternator" is obtained. If the field cut by the straight conductor is
of uniform density, and all the lines pass straight across from one
pole-face to the other (both of which assumptions are approximately
correct), a curve connecting the instantaneous values of the E.M.F. as
ordinates with time or degrees of angular movement as abscissae (as
shown at the foot of fig. 3), will, if the speed of rotation be uniform,
be a sine curve. If, however, the conductor is mounted on an iron
cylinder (fig. 4),[8] a sufficient margin being allowed for mechanical
clearance between it and the poles, not only will the reluctance of the
magnetic circuit be reduced and the total flux and its density in the
air-gap B_g be thereby increased, but the path of the lines will become
nearly radial, except at the "fringe" near the edges of the pole-tips;
hence the relative directions of the movement and of the lines will be
continuously at right angles. The shape of the E.M.F. curve will then be
as shown in fig. 4--flat-topped, with rounded corners rapidly sloping
down to the zero line.
[Illustration: FIG. 4.]
But a single wire cannot thus be made to give more than a few volts, and
while dynamos for voltages from 5 to 10 are required for certain
purposes, the voltages in common use range from 100 to 10,000. It is
therefore necessary to connect a number of such wires in series, so as
to form an "armature winding." If several similar conductors are
arranged along the length of the iron core parallel to the first (fig.
5), the E.M.F.'s generated in the conductors which at any moment are
under the same pole are similarly directed, and are opposite to the
directions of the E.M.F.'s in the conductors under the other pole (cf
fig. 5 where the dotted and crossed ends of the wires indicate E.M.F.'s
directed respectively towards and away from the observer). Two distinct
methods of winding thence arise, the similarity of the E.M.F.'s under
the same pole being taken advantage of in the first, and the opposite
E.M.F.'s under N and S poles in the second.
[Illustration: FIG. 5.]
[Illustration: FIG. 6.]
1. The first, or _ring_-winding, was invented by Dr Antonio Pacinotti of
Florence[9] in 1860, and was subsequently and independently reintroduced
in 1870[10] by the Belgian electrician, Zenobe Theophile Gramme, whence
it is also frequently called the "Gramme" winding. By this method the
farther end of conductor 1 (fig. 5) is joined in series to the near end
of conductor 2; this latter lies next to it on the surface of the core
or immediately above it, so that both are simultaneously under the same
pole-piece. For this series connexion to be possible, the armature core
must be a hollow cylinder, supported from the shaft on an open
non-magnetic spider or hub, between the arms of which there is room for
the internal wire completing the loop (fig. 6). The end of one complete
loop or turn embracing one side of the armature core thus forms the
starting-point for another loop, and the process can be continued if
required to form a coil of two or more turns. In the ring armature the
iron core serves the double purpose of conducting the lines across from
one pole to the other, and also of shielding from the magnetic flux the
hollow interior through which the connecting wires pass. Any lines which
leak across the central space are cut by the internal wires, and the
direction of cutting is such that the E.M.F. caused thereby opposes the
E.M.F. due to the active conductors proper on the external surface. If,
however, the section of iron in the core be correctly proportioned, the
number of lines which cross the interior will bear but a small ratio to
those which pass entirely through the iron, and the counter E.M.F. of
the internal wires will become very small; they may then be regarded
simply as connectors for joining the external active wires in series.
[Illustration: FIG. 7.]
2. The second or _drum_ method was used in the original "shuttle-wound"
armatures invented by Dr Werner von Siemens in 1856, and is sometimes
called the "Siemens" winding. The farther end of conductor 1 (fig. 5) is
joined by a connecting wire to the farther end of another conductor 2'
situated nearly diametrically opposite on the other side of the core and
under the opposite pole-piece. The near end of the complete loop or turn
is then brought across the end of the core, and can be used as the
starting-point for another loop beginning with conductor 2, which is
situated by the side of the first conductor. The iron core may now be
solid from the surface to the shaft, since no connecting wires are
brought through the centre, and each loop embraces the entire armature
core (fig. 7). By the formation of two loops in the ring armature and of
the single loop in the drum armature, two active wires are placed in
series; the curves of instantaneous E.M.F. are therefore similar in
shape to that of the single wire (fig. 4), but with their ordinates
raised throughout to double their former height, as shown at the foot of
fig. 6.
Next, if the free ends of either the ring or drum loops, instead of
being connected to two collecting rings, are attached to the two halves
of a split-ring insulated from the shaft (as shown in fig. 7 in
connexion with a drum armature), and the stationary brushes are so set
relatively to the loops that they pass over from the one half of the
split-ring to the other half at the moment when the loops are passing
the centre of the interpolar gap, and so are giving little or no E.M.F.,
each brush will always remain either positive or negative. The current
in the external circuit attached to the brushes will then have a
constant direction, although the E.M.F. in the active wires still
remains alternating; the curve of E.M.F. obtained at the brushes is thus
(as in fig. 7) entirely above the zero line. The first dynamo of H.
Pixii,[11] which immediately followed Faraday's discovery, gave an
alternating current, but in 1832[12] the alternator was converted into a
machine giving a _unidirected current_ by the substitution of a
rudimentary "commutator" in place of mercury collecting cups.
(B) So far the length of the active wires has been parallel to the axis
of rotation, but they may equally well be arranged perpendicularly
thereto. The poles will then have plane faces and the active wires will
be disposed with their length approximately radial to the axis of the
shaft. In order to add their E.M.F.'s in series, two types of winding
may be employed, which are precisely analogous in principle to the ring
and drum windings under arrangement (A).
3. The _discoidal_ or flat-ring armature is equivalent to a ring of
which the radial depth greatly exceeds the length, with the poles
presented to one side of the ring instead of embracing its cylindrical
surface. A similar set of poles is also presented to the opposite side
of the ring, like poles being opposite to one another, so that in effect
each polar surface is divided into two halves, and the groups of lines
from each side bifurcate and pass circumferentially through the armature
core to issue into the adjacent poles of opposite sign.
4. In the _disk_ machine, no iron core is necessary for the armature,
the two opposite poles of unlike sign being brought close together,
leaving but a short path for the lines in the air-gap through which the
active wires are rotated.
If the above elementary dynamos are compared with fig. 1, it will be
found that they all possess a distinctive feature which is not present
in the original disk machine of Faraday. In the four types of machine
above described each active wire in each revolution first cuts the group
of lines forming a field in one direction, and then cuts the same lines
again in the opposite direction relatively to the sense of the lines, so
that along the length of the wire the E.M.F. alternates in direction.
But in the dynamo of fig. 1 the sector of the copper disk which is at
any moment moving through the magnetic field and which forms the single
active element is always cutting the lines in the same manner, so that
the E.M.F. generated along its radial length is continuous and unchanged
in direction. This radical distinction differentiates the two classes of
_heteropolar_ and _homopolar_ dynamos, Faraday's disk machine of fig. 1
being the type of the latter class. In it the active element may be
arranged either parallel or at right angles to the axis of rotation;
but in both cases, in order to increase the E.M.F. by placing two or
more elements in series, it becomes necessary either (1) to employ some
form of sliding contact by which the current may be collected from the
end of one active element and passed round a connecting wire into the
next element without again cutting the field in the reverse direction,
or (2) to form on the armature a loop of which each side is alternately
active and inactive. The first method limits the possibilities of the
homopolar machine so greatly when large currents and high voltages are
required that it is now only used in rare instances, as e.g.
occasionally in dynamos driven by steam-turbines which have a very high
speed of rotation. The second alternative may be carried into effect
with any of the four methods of armature winding, but is practically
confined to the drum and disk types. In its drum form the field is
divided into two or more projecting poles, all of the same sign, with
intervening neutral spaces of equal width, and the span of the loop in
the direction of rotation is at least equal to the width of a polar
projection, as in fig. 8, where two polar projections are shown. Each
side of the loop then plays a dual part; it first cuts the lines of one
polar projection and generates an E.M.F., and next becomes an inactive
connecting wire, while the action is taken up by the opposite side of
the loop which has previously served as a connector but now cuts the
lines of the next polar projection. The E.M.F. is thus always in the
same direction along the side which is at any moment active, but
alternates round the loop as a whole, and the distinctive peculiarity of
the homopolar machine, so soon as any form of "winding" is introduced
into its armature, is lost. It results that the homopolar principle,
which would prima facie appear specially suitable for the generation of
a unidirectional E.M.F. and continuous current, can seldom be used for
this purpose and is practically confined to alternators. It may
therefore be said that in almost all dynamos, whether they supply an
alternating or a continuous current in the external circuit, the E.M.F.
and current in the armature are alternating.
[Illustration: FIG. 8.]
Ring winding was largely employed in early continuous-current dynamos
and also in the alternators of Gramme and H. Wilde, and later of Auguste
de Meritens. Disk winding was also successfully introduced for
alternators, as in the magneto-machines of Nollet (1849) and the
alternators of Wilde (1866) and Siemens (1878), and its use was
continued in the machines of W.M. Mordey and S.Z. Ferranti. But although
the ring, discoidal-ring and disk methods of winding deserve mention
from their historical importance, experience has shown that drum winding
possesses a marked superiority for both electrical and manufacturing
reasons; the three former methods have in fact been practically
discarded in its favour, so that the drum method will hereafter alone be
considered.
The drum coil, composed of several loops wound side by side, may
therefore be regarded as the constituent active element out of which the
armature winding of the modern dynamo is developed. Its application to
the multipolar machine is easily followed from fig. 9, which illustrates
the heteropolar type of dynamo. The span of the loops, which is nearly
180 deg. or across the diameter of the two-pole machine, is reduced
approximately to 90 deg. in the four-pole or to 60 deg. in the six-pole machine
and so on, the curvature of the coil becoming gradually less as the
number of poles is increased. The passage of a coil through two magnetic
fields of opposite direction yields a complete wave of E.M.F., such as
is shown in fig. 6, and the time in seconds taken to pass through such a
complete cycle is the "period" of the alternating E.M.F. The number of
complete periods through which the E.M.F. of the coil passes per second
is called the "periodicity" or "frequency" of the machine. In the
bipolar machine this is equal to the number of revolutions per second,
and in the multipolar machine it is equal to the number of pairs of
fields through which the coil passes in one second; hence in general the
periodicity is pN/60, where N = the number of revolutions per minute and
p = the number of pairs of poles, and this holds true of the E.M.F. and
current round the coil, even though the E.M.F. and current furnished to
the external circuit may be rendered unidirectional or continuous. The
only difference on this point is that in the continuous-current machine
the poles are usually fewer than in the alternator, and the periodicity
is correspondingly lower. Thus in the former case the number of poles
ranges from 2 to 12 and the usual frequencies from 5 to 20; but with
alternators the frequencies in commercial use range from 25 to 120, and
in large machines driven by slow-speed engines the number of poles may
even be as high as 96.
[Illustration: FIG. 9.
I. Smooth. II. Toothed.]
[Illustration: FIG. 10.]
The drum coil may be applied either to the external surface of a
rotating armature, the field-magnet being external and stationary (fig.
9), or to the internal surface of a stationary armature (fig. 10), the
field-magnet being internal and rotating. While the former combination
is universally adopted in the continuous-current dynamo, the latter is
more usual in the modern alternator. In either case the iron armature
core must be "laminated"; the passage of the lines of the field across
its surface sets up E.M.F.'s which are in opposite directions under
poles of opposite sign, so that if the core were a solid mass a
current-sheet would flow along its surface opposite to a pole, and
complete its circuit by passing through the deeper layers of metal or by
returning in a sheet under a pole of opposite sign. Such "eddy-currents"
can be practically avoided by dividing the metal core into laminations
at right angles to the length of the active wires which are themselves
arranged to secure the greatest rate of line-cutting and maximum E.M.F.
The production of the eddy-current E.M.F. is not thereby prevented, but
the paths of the eddy-currents are so broken up that the comparatively
high resistance with which they meet reduces their amount very greatly.
The laminae must be lightly insulated from one another, right up to
their edges, so that the E.M.F.'s which still act across their thickness
will not be added up along the length of the core, but will only produce
extremely small currents circulating through the interior of the
separate laminations. Each thin iron plate is either coated with an
insulating varnish or has one of its sides covered with a sheet of very
thin paper; the thickness of the laminae is usually about one-fortieth
of an inch, and if this is not exceeded the rate at which energy is
dissipated by eddy-currents in the core is so far reduced that it does
not seriously impair the efficiency of the machine.
Lastly, the drum coils may be either attached to the surface of a smooth
armature core (fig. 9, I.), or may be wound through holes formed close
to the periphery of the core, or may be embedded in the slots between
projecting iron teeth (figs. 9 [II.] and 10). Originally employed by
Antonio Pacinotti in connexion with ring winding, the toothed armature
was after some considerable use largely discarded in favour of the
smooth core; it has, however, been reintroduced with a fuller
understanding of the special precautions necessitated in its design, and
it is now so commonly used that it may be said to have superseded the
smooth-surface armature.
Not only does the toothed armature reduce the length of the air-gap to
the minimum permitted by mechanical and magnetic considerations, and
furnish better mechanical protection to the armature coils, but it
also ensures the positive holding of the active wires against the
mechanical drag which they experience as they pass through the
magnetic field. Further, the active wires in the toothed armature are
relieved of a large proportion of this mechanical drag, which is
transferred to the iron teeth. The lines of the field, after passing
through the air-gap proper, divide between the teeth and the slots in
proportion to their relative permeances. Hence at any moment the
active wires are situated in a weak field, and for a given armature
current the force on them is only proportional to this weak field.
This important result is connected with the fact that when the
armature is giving current the distribution of the lines over the face
of each tooth is distorted, so that they become denser on the
"trailing" side than on the "leading" side;[13] the effect of the
non-uniform distribution acting on all the teeth is to produce a
magnetic drag on the armature core proportional to the current passing
through the wires, so that the total resisting force remains the same
as if the armature had a smooth core. The amount by which the stress
on the active wires is reduced entirely depends upon the degree to
which the teeth are saturated, but, since the relative permeability of
iron even at a flux density of 20,000 lines per sq. cm. is to that of
air approximately as 33:1, the embedded wires are very largely
relieved of the driving stress. An additional gain is that solid bars
of much greater width can be used in the toothed armature than on a
smooth core without appreciable loss from eddy-currents within their
mass.
A disadvantage of the slotted core is, however, that it usually
necessitates the lamination of the pole-pieces. If the top of the slot
is open, and its width of opening is considerably greater than the
length of the air-gap from the iron of the pole-face to the surface of
the teeth, the lines become unequally distributed not only at the
surface of the teeth, but also at the face of the pole-pieces; and
this massing of the lines into bands causes the density at the
pole-face to be rhythmically varied as the teeth pass under it. No
such variation can take place in a solid mass of metal without the
production of eddy-currents within it; hence if the width of the
slot-opening is equal to or exceeds twice the length of the single
air-gap, lamination of the pole-pieces in the same plane as that of
the armature core becomes advisable.
If the wires are threaded through holes or tunnels pierced close to
the periphery of the core, the same advantages are gained as with open
slots, and lamination of the pole-pieces is rendered unnecessary. But
on the other hand, the process of winding becomes laborious and
expensive, while the increase in the inductance of the coils owing to
their being surrounded by a closed iron circuit is prejudicial to
sparkless commutation in the continuous-current dynamo and to the
regulation of the voltage of the alternator. A compromise is found in
the half-closed slot, which is not uncommon in alternators, although
the open slot is more usual in continuous-current dynamos.
With the addition of more turns to the elementary drum loop or of
several complete coils, new questions arise, and in connexion therewith
the two great classes of machines, viz. alternators and
continuous-current dynamos, which have above been treated side by side,
diverge considerably, so that they are best considered separately. The
electromotive-force equation of the alternator will be first deduced,
and subsequently that of the continuous-current machine.
[Illustration: FIG. 11.]
Corresponding to the number of pairs of poles in the multipolar
alternator, it is evident that there may also be an equal number of
coils as shown diagrammatically in fig. 11. The additional coils, being
similarly situated in respect to other pairs of poles, will exactly
reproduce the E.M.F. of the original coil in phase and magnitude, so
that when they are connected in series the total E.M.F. will be
proportional to the number of coils in series; or if they are connected
in parallel, while not adding to the E.M.F., they will proportionately
increase the current-carrying capacity of the combination. But within
each coil the addition of more loops will not cause an equal increase in
the total E.M.F., unless the phases of the component E.M.F.'s due to the
several turns are identical, and on this account it becomes necessary to
consider the effect of the width of the coil-side.
[Illustration: FIG. 12.]
If the additional loops are wound within the same slots as the original
loop, the winding is "concentrated," and each turn will then add the
same E.M.F. But if the coil-side is divided between two or more slots,
the phase of the E.M.F. yielded by the wires in one slot being different
from that of the wires in another neighbouring slot, the sum of all the
E.M.F.'s will be less than the E.M.F. of one component loop multiplied
by the number of loops or turns in the coil. The percentage reduction in
the E.M.F. will depend upon the number of the slots in a coil-side and
their distance apart, i.e. on the virtual width of the coil-side
expressed as a fraction of the "pole-pitch" or the distance measured
along the pitch-line from the centre of one pole to the centre of a
neighbouring pole of opposite sign (fig. 12). The winding is now to be
regarded as "grouped," since a small number of distinct phases
corresponding to the groups within the two, three or four slots have to
be compounded together. As the number of slots per coil-side is
increased, an approach is gradually made to the case of "uniform
distribution," such as would obtain in a smooth-core armature in which
the turns of the coil are wound closely side by side. Thus in the
six-turn coil of fig. 12 A, which represents the development of a
two-pole armature when the core is cut down to the shaft and opened out
flat, there are in effect six phases compounded together, each of which
differs but little from that of its next neighbour. With numerous wires
lying still closer together a large number of phases are compounded
until the distribution becomes practically uniform; the decrease in the
E.M.F., as compared with that of a single turn multiplied by the number
in series, is then immediately dependent upon the width of the coil-side
relatively to the pole-pitch.
[Illustration: FIG. 13.]
If the width of the inner loop of fig. 12 A is less than that of the
pole-face, its two sides will for some portion of each period be moving
under the same pole, and "differential action" results, the net E.M.F.
being only that due to the difference between the E.M.F.'s of the two
sides. The loop of smallest width must therefore exceed the width of
pole-face, if direct differential action is to be avoided. The same
consideration also determines the width of the outer loop; if this be
deducted from twice the pole-pitch, the difference should not be less
than the width of the pole-face, so that, e.g., in a bipolar machine the
outer loop may stand to the S. pole exactly as the inner loop stands to
the N. pole (fig. 13). In other words, the width of the coil-side must
not exceed the width of the interpolar gap between two fields. Evidently
then if the ratio of the pole-width to the pole-pitch approaches unity,
the width of the coil-side must be very small, and vice versa. A
compromise between these conflicting considerations is found if the pole
is made not much more than half the pole-pitch, and the width of the
coil-side is similarly about half the pole-pitch and therefore equal in
width to the pole (fig. 13). A single large coil, such as that of fig.
12 A, can, however, equally well be divided into two halves by taking
the end-connexions of one half of the turns round the opposite side of
the shaft (fig. 12 B), as indeed has already been done in fig. 13. Each
sheaf or band of active wires corresponding to a pole is thereby
unaffected, but the advantages are gained that the axial length of the
end-connexions is halved, and that they have less inductance. Thus if in
fig. 11 there are four turns per coil, fig. 14 is electrically
equivalent to it (save that the coils are here shown divided into two
parallel paths, each carrying half the total current). When the large
coils are divided as above described, it results that there are as many
coils as there are poles, the outer loop of the small coil having a
width equal to the pole-pitch, and the inner a width equal to the
pole-face.
[Illustration: FIG. 14.]
Such is the form which the "single-phase alternator" takes, but since
only one-half of the armature core is now covered with winding, an
entirely distinct but similar set of coils may be wound to form a second
armature circuit between the coils of the first circuit. The phase of
this second circuit will differ by 90 deg. or a quarter of a period from
that of the first, and it may either be used to feed an entirely
separate external circuit possibly at a different pressure or, if it be
composed of the same number of turns and therefore gives the same
voltage, it may be interconnected with the first circuit to form a
"quarter-phase alternator," as will be more fully described later. By an
extension of the same process, if the width of each side of a coil is
reduced to one-sixth of the pole-pitch, three armature circuits can be
wound on the same core, and a "three-phase alternator," giving waves of
E.M.F. differing in phase by 120 deg., is obtained.
The fundamental "electromotive-force equation" of the heteropolar
alternator can now be given a more definite form. Let Z_a be the
number of C. G. S. lines or the total flux, which issuing from any
one pole flows through the armature core, to leave it by another pole
of opposite sign. Since each active wire cuts these lines, first as
they enter the armature core and then as they emerge from it to enter
another pole, the total number of lines cut in one revolution by any
one active wire is 2pZ_a. The time in seconds taken by one revolution
is 60/N. The average E.M.F. induced in each active wire in one
revolution being proportional to the number of lines cut divided by
the time taken to cut them is therefore 2Z_a(pN/60) x 10^(-8) volts.
The active wires which are in series and form one distinct phase may
be divided into as many bands as there are poles; let each such band
contain t active wires, which as before explained may either form one
side of a single large coil or the adjacent sides of two coils when
the large coil is divided into two halves. Since the wires are joined
up into loops, two bands are best considered together, which with
either arrangement yield in effect a single coil of t turns. The
average E.M.F.'s of all the wires in the two bands when added together
will therefore be 4Z_a(pN/60)t x 10^(-8). But unless each band is
concentrated within a single slot, there must be some differential
action as they cross the neutral line between the poles, so that the
last expression is virtually the _gross_ average E.M.F. of the loops
on the assumption that the component E.M.F.'s always act in agreement
round the coil and do not at times partially neutralize one another.
The _net_ average E.M.F. of the coil as a whole, or the arithmetical
mean of all the instantaneous values of a half-wave of the actual
E.M.F. curve, is therefore reduced to an extent depending upon the
amount of differential action and so upon the width of the coil-side
when this is not concentrated. Let k' = the coefficient by which the
gross average E.M.F. must be multiplied to give the net average
E.M.F.; then k' may be called the "width-factor," and will have some
value less than unity when the wires of each band are spread over a
number of slots. The net average E.M.F. of the two bands corresponding
to a pair of poles is thus e_(av) = 4k'Z_a(pN/60)t x 10^(-8).
The shape of the curve of instantaneous E.M.F. of the coil must
further be taken into account. The "effective" value of an alternating
E.M.F. is equal to the square root of the mean square of its
instantaneous values, since this is the value of the equivalent
unidirectional and unvarying E.M.F., which when applied to a given
resistance develops energy at the same rate as the alternating E.M.F.,
when the effect of the latter is averaged over one or any whole number
of periods. Let k" = the ratio of the square root of the mean square
to the average E.M.F. of the coil, i.e. = effective E.M.F./average
E.M.F. Since it depends upon the shape of the E.M.F. curve, k" is also
known as the "form-factor"; thus if the length of gap between
pole-face and armature core and the spacing of the wires were so
graduated as to give a curve of E.M.F. varying after a sine law, the
form-factor would have the particular value of [pi]/2[root]2 = 1.11,
and to this condition practical alternators more or less conform. The
effective E.M.F. of the two bands corresponding to a pair of poles is
thus e_(eff) = 4k'k"Z_a(pN/60)t x 10^(-8).
In any one phase there are p pairs of bands, and these may be divided
into q parallel paths, where q is one or any whole number of which p
is a multiple. The effective E.M.F. of a complete phase is therefore
pe{eff}/q. Lastly, if m = the number of phases into which the armature
winding is divided, and [tau] = the total number of active wires on
the armature counted all round its periphery, t = [tau]/2pm, and the
effective E.M.F. per phase is E_a = 2k'k"Z_a(pN[tau]/60mq) x 10^(-8).
The two factors k' and k" may be united into one coefficient, and the
equation then takes its final form
E_a = 2KZ_a(pN[tau]/60mq) x 10^(-8) volts (1a)
In the alternator q is most commonly 1, and there is only one circuit
per phase; finally the value of K or the product of the width-factor
and the form-factor usually falls between the limits of 1 and 1.25.
We have next to consider the effect of the addition of more armature
loops in the case of dynamos which give a unidirectional E.M.F. in
virtue of their split-ring collecting device, i.e. of the type shown in
fig. 7 with drum armature or its equivalent ring form. As before, if the
additional loops are wound in continuation of the first as one coil
connected to a single split-ring, this coil must be more or less
concentrated into a narrow band; since if the width becomes nearly equal
to or exceeds the width of the interpolar gap, the two edges of the
coil-side will just as in the alternator act differentially against one
another during part of each revolution. The drum winding with a single
coil thus gives an armature of the H- or "shuttle" form invented by Dr
Werner von Siemens. Although the E.M.F. of such an arrangement may have
a much higher maximum value than that of the curve of fig. 7 for a
single loop, yet it still periodically varies during each revolution and
so gives a pulsating current, which is for most practical uses
unsuitable. But such pulsation might be largely reduced if, for example,
a second coil were placed at right angles to the original coil and the
two were connected in series; the crests of the wave of E.M.F. of the
second coil will then coincide with the hollows of the first wave,
and although the maximum of the resultant curve of E.M.F. may be no
higher its fluctuations will be greatly decreased. A spacial
displacement of the new coils along the pole-pitch, somewhat as in a
polyphase machine, thus suggests itself, and the process may be carried
still further by increasing the number of equally spaced coils, provided
that they can be connected in series and yet can have their connexion
with the external circuit reversed as they pass the neutral line between
the poles.
[Illustration: FIG. 15.]
[Illustration: FIG. 16.]
Given two coils at right angles and with their split-rings displaced
through a corresponding angle of 90 deg., they may be connected in series by
joining one brush to the opposite brush of the second coil, the external
circuit being applied to the two remaining brushes.[14] The same
arrangement may again be repeated with another pair of coils in parallel
with the first, and we thus obtain fig. 15 with four split-rings, their
connexions to the loops being marked by corresponding numerals; the four
coils will give the same E.M.F. as the two, but they will be jointly
capable of carrying twice the current, owing to their division into two
parallel circuits. Now in place of the four split-rings may be employed
the greatly simplified four-segment structure shown in fig. 16, which
serves precisely the same purpose as the four split-rings but only
requires two instead of eight brushes. The effect of joining brush 2 in
fig. 15 across to brush 3, brush 4 to brush 5, 5 to 6, &c., has
virtually been to connect the end of coil A with the beginning of coil
B, and the end of coil B with the beginning of coil A', and so on, until
they form a continuous closed helix. Each sector of fig. 16 will
therefore replace two halves of a pair of adjacent split-rings, if the
end and beginning of a pair of adjacent coils are connected to it in a
regular order of sequence. The four sectors are insulated from one
another and from the shaft, and the whole structure is known as the
"commutator,"[15] its function being not simply to collect the current
but also to commute its direction in any coil as it passes the
interpolar gap. The principle of the "closed-coil continuous-current
armature" is thus reached, in which there are at least two parallel
circuits from brush to brush, and from which a practically steady
current can be obtained. Each coil is successively short-circuited, as a
brush bridges over the insulation between the two sectors which
terminate it; and the brushes must be so set that the period of
short-circuit takes place when the coil is generating little or no
E.M.F., i.e. when it is moving through the zone between the pole-tips.
The effect of the four coils in reducing the percentage fluctuation of
the E.M.F. is very marked, as shown at the foot of fig. 15 (where the
upper curve is the resultant obtained by adding together the separate
curves of coils A and B), and the levelling process may evidently be
carried still further by the insertion of more coils and more
corresponding sectors in the commutator, until the whole armature is
covered with winding. For example, figs. 17 and 18 show a ring and a
drum armature, each with eight coils and eight commutator sectors; their
resultant curve, on the assumption that a single active wire gives the
flat-topped curve of fig. 4, will be the upper wavy line of E.M.F.
obtained by adding together two of the resultant curves of fig. 15, with
a relative displacement of 45 deg.. The amount of fluctuation for a given
number of commutator sectors depends upon the shape of the curve of
E.M.F. yielded by the separate small sections of the armature winding;
the greater the polar arc, the less the fluctuation. In practice, with a
polar arc equal to about 0.75 of the pitch, any number of sectors over
32 per pair of poles yields an E.M.F. which is sensibly constant
throughout one or any number of revolutions.
[Illustration: FIG. 17.]
[Illustration: FIG. 18.]
The fundamental electro-motive-force equation of the
continuous-current heteropolar machine is easily obtained by analogy
from that of the alternator. The gross average E.W.F. from the two
sides of a drum loop without reference to its direction is as before
4Z_a(pN/60) x 10^(-8) volts. But for two reasons its net average
E.M.F. may be less; the span of the loop may be less than the
pole-pitch, so that even when the brushes are so set that the position
of short-circuit falls on the line where the field changes its
direction, the two sides of the loop for some little time act against
each other; or, secondly, even if the span of the loop be equal to the
pole-pitch, the brushes may be so set that the reversal of the
direction of its induced E.M.F. does not coincide with reversal of the
current by the passage of the coil under the brushes. The net average
E.M.F. of the loop is therefore proportional to the algebraic sum of
the lines which it cuts in passing from one brush to another, and this
is equal to the net amount of the flux which is included within the
loop when situated in the position of short-circuit under a brush. The
amount of this flux may be expressed as k'Z_a where k' is some
coefficient, less than unity if the span of the coil be less than the
pole-pitch, and also varying with the position of the brushes. The net
average E.M.F. of the loop is therefore
4k'Z_a(pN/60) x 10^(-8).
In practice the number of sections of the armature winding is so large
and their distribution round the armature periphery is so uniform,
that the sum total of the instantaneous E.M.F.'s of the several
sections which are in series becomes at any moment equal to the net
average E.M.F. of one loop multiplied by the number which are in
series. If the winding is divided into q parallel circuits, the number
of loops in series is [tau]/2q, so that the total E.M.F. is E_a =
2(k'/q)Z_a(pN/60)[tau] x 10^(-8) volts. Thus as compared with the
alternator not only is there no division of the winding into separate
phases, but the form-factor k' disappears, since the effective and
average E.M.F.'s are the same. Further whereas in the alternator q may
= 1, in the continuous-current closed-coil armature there can never be
less than two circuits in parallel from brush to brush, and if more,
their number must always be a multiple of two, so that q can never be
less than two and must always be an even number. Lastly, the factor k'
is usually so closely equal to 1, that the simplified equation may in
practice be adopted, viz.
E_a = (2/q)(ZpN/60)[tau] x 10^(-8) volts (1b)
The fundamental equation of the electromotive force of the dynamo in
its fully developed forms (1 a) (and 1 b) may be compared with its
previous simple statement (1.). The three variable terms still find
their equivalents, but are differently expressed, the density B_g
being replaced by the total flux of one field Z_a, the length L of the
single active wire by the total number of such wires [tau], and the
velocity of movement V by the number of revolutions per second. Even
when the speed is fixed, an endless number of changes may be rung by
altering the relative values of the remaining two factors; and in
successful practice these may be varied between fairly wide limits
without detriment to the working or economy of the machine. While it
may be said that the equation of the E.M.F. was implicitly known from
Faraday's time onwards, the difficulty under which designers laboured
in early days was the problem of choosing the correct relation of Z_a
or [tau] for the required output; this, again, was due chiefly to the
difficulty of predetermining the total flux before the machine was
constructed. The general error lay in employing too weak a field and
too many turns on the armature, and credit must here be given to the
American inventors, E. Weston and T.A. Edison, for their early
appreciation of the superiority in practical working of the drum
armature, with comparatively few active wires rotating in a strong
field.
The armature core.
_Continuous-current Dynamos._--On passing to the separate consideration
of alternators and continuous-current dynamos, the chief constructive
features of the latter will first be taken in greater detail. As already
stated in the continuous-current dynamo the armature is usually the
rotating portion, and the necessity of laminating its core has been
generally described. The thin iron stampings employed to build up the
core take the form of circular washers or "disks," which in small
machines are strung directly on the shaft; in larger multipolar
machines, in which the required radial depth of iron is small relatively
to the diameter, a central cast iron hub supports the disks. Since the
driving force is transmitted through the shaft to the disks, they must
in the former case be securely fixed by keys sunk into the shaft; when a
central hub is employed (fig. 19) it is keyed to the shaft, and its
projecting arms engage in notches stamped on the inner circumference of
the disks, or the latter have dovetailed projections fitting into the
arms. The disks are then tightly compressed and clamped between stout
end-plates so as to form a nearly solid iron cylinder of axial length
slightly exceeding the corresponding dimension of the poles. If the
armature is more than 4 ft. in diameter, the disks become too large to
be conveniently handled in one piece, and are therefore made in
segments, which are built up so as to break joint alternately. Prior to
assemblage, the external circumference of each disk is notched in a
stamping machine with the required number of slots to receive the
armature coils, and the longitudinal grooves thereby formed in the
finished core only require to have their sharp edges smoothed off so
that there may be no risk of injury to the insulation of the coils.
[Illustration: FIG. 19.]
Armature winding.
With open slots either the armature coils may be encased with wrappings
of oiled linen, varnished paper and thin flexible micanite sheeting in
order to insulate them electrically from the iron slots in which they
are afterwards embedded; or the slots may be themselves lined with
moulded troughs of micanite, &c., for the reception of the armature
coils, the latter method being necessary with half-closed slots.
According to the nature of the coils armatures may be divided into the
two classes of coil-wound and bar-wound. In the former class, round
copper wire, double-cotton covered, is employed, and the coils are
either wound by hand directly on to the armature core, or are shaped on
formers prior to being inserted in the armature slots. Hand-winding is
now only employed in very small bipolar machines, the process being
expensive and accompanied by the disadvantage that if one section
requires to be repaired, the whole armature usually has to be dismantled
and re-wound. Former-wound coils are, on the other hand, economical in
labour, perfectly symmetrical and interchangeable, and can be thoroughly
insulated before they are placed in the slots. The shapers employed in
the forming process are very various, but are usually arranged to give
to the finished coil a lozenge shape, the two straight active sides
which fit into the straight slots being joined by V-shaped ends; at each
apex of the coil the wire is given a twist, so that the two sides fall
into different levels, an upper and a lower, corresponding to the two
layers which the coil-sides form on the finished armature. Rectangular
wire of comparatively small section may be similarly treated, and if
only one loop is required per section, wide and thin strip can be bent
into a complete loop, so that the only soldered joints are those at the
commutator end where the loops are interconnected. But finally with
massive rectangular conductors, the transition must be made to
bar-winding, in which each bar is a half-loop, insulated by being taped
after it has been bent to the required shape; the separate bars are
arranged on the armature in two layers, and their ends are soldered
together subsequently to form loops. As a general rule, whether bars or
former-wound coils are employed, the armature is barrel-wound, i.e. the
end-connexions project outwards from the slots with but little change of
level, so that they form a cylindrical mass supported on projections
from the end-plates of the core (fig. 19); but, in certain cases, the
end-connexions are bent downwards at right angles to the shaft, and they
may then consist of separate strips of copper bent to a so-called
butterfly or evolute shape.
After the coils or loops have been assembled in the slots on the
armature core, and the commutator has been fixed in place on the shaft,
the soldering of the ends of the coils proceeds, by which at once the
union of the end of one coil with the beginning of the next, and also
their connexion to the commutator sectors, is effected, and in this lies
the essential part of armature winding.
[Illustration: FIG. 20.
Lap-loops]
Lap-winding.
The development of the modern drum armature, with its numerous coils
connected in orderly sequence into a symmetrical winding, as
contrasted with the earlier Siemens armatures, was initiated by F. von
Hefner Alteneck (1871), and the laws governing the interconnexion of
the coils have now been elaborated into a definite system of winding
formulae. Whatever the number of wires or bars in each side of a coil,
i.e. whether it consist of a single loop or of many turns, the final
connexions of its free ends are not thereby affected, and it may be
mentally replaced by a single loop with two active inducing sides. The
coil-sides in their final position are thus to be regarded as separate
primary elements, even in number, and distributed uniformly round the
armature periphery or divided into small, equally spaced groups by
being located within the slots of a toothed armature. Attention must
then be directed simply to the span of the back connexion between the
elements at the end of the armature further from the commutator, and
to the span of the front connexion by which the last turn of a coil is
finally connected to the first turn of the next in sequence, precisely
as if each coil of many turns were reduced to a single loop. In order
to avoid direct differential action, the span of the back connexion
which fixes the width of the coil must exceed the width of the
pole-face, and should not be far different from the pole-pitch; it is
usually a little less than the pole-pitch. Taking any one element as
No. 1 in fig. 20, where for simplicity a smooth-core bipolar armature
is shown, the number of winding-spaces, each to be occupied by an
element, which must be counted off in order to find the position of
the next element in series, is called the "pitch" of the
end-connexion, front or back, as the case may be. Thus the back pitch
of the winding as marked by the dotted line in fig. 20 is 7, the
second side of the first loop being the element numbered 1 + 7 = 8. In
forming the front end-connexion which completes the loop and joins it
to the next in succession, two possible cases present themselves. By
the first, or "lap-winding," the front end-connexion is brought
backwards, and passing on its way to a junction with a commutator
sector is led to a third element lying within the two sides of the
first loop, i.e. the second loop starts with the element, No. 3, lying
next but one to the starting-point of the first loop. The winding
therefore returns backwards on itself to form each front end, but as a
whole it works continually forwards round the armature, until it
finally "re-enters," after every element has been traversed. The
development of the completed winding on a flat surface shows that it
takes the form of a number of partially overlapping loops, whence its
name originates. The firm-line portion of fig. 21 gives the
development of an armature similar to that of fig. 18 when cut through
at the point marked X and opened out; two of the overlapping loops are
marked thereon in heavy lines. The multipolar lap-wound armature is
obtained by simply repeating the bipolar winding p times, as indicated
by the dotted additions of fig. 21 which convert it from a two-pole to
a four-pole machine. The characteristic feature of the lap-wound
armature is that there are as many parallel paths from brush to brush,
and as many points at which the current must be collected, as there
are poles. As the bipolar closed-coil continuous-current armature has
been shown to consist in reality of two circuits in parallel, each
giving the same E.M.F. and carrying half the total current, so the
multipolar lap-wound drum consists of p pairs of parallel paths, each
giving the same E.M.F. and carrying 1/2p of the total current. Thus in
equation 1.b we have q = 2p, and the special form which the _E.M.F.
equation of the lap-wound armature_ takes is E_[alpha] = Z_a
(N/60)[tau] x 10^(-8) volts. All the brushes which are of the same
sign must be connected together in order to collect the total armature
current. The several brush-sets of the multipolar lap-wound machine
may again be reduced to two by "cross-connexion" of sectors situated
360 deg./p apart, but this is seldom done, since the commutator must then
be lengthened p times in order to obtain the necessary brush
contact-surface for the collection of the entire current.
[Illustration: FIG. 21.]
[Illustration: FIG. 22.
Wave-loops]
[Illustration: FIG. 23.]
Wave-winding.
But for many purposes, especially where the voltage is high and the
current small, it is advantageous to add together the inductive effect
of the several poles of the multipolar machine by throwing the E.M.F's
of half the total number of elements into series, the number of
parallel circuits being conversely again reduced to two. This is
effected by the second method of winding the closed-coil continuous
current drum, which is known as "wave-winding." The front pitch is now
in the same direction round the armature as the back pitch (fig. 22),
so that the beginning of the second loop, i.e. element No. 15, lies
outside the first loop. After p loops have been formed and as many
elements have been traversed as there are poles, the distance covered
either falls short of or exceeds a complete tour of the armature by
two winding-spaces, or the width of two elements. A second and third
tour are then made, and so on, until finally the winding again closes
upon itself. When the completed winding is developed as in fig. 23, it
is seen to work continuously forwards round the armature in zigzag
waves, one of which is marked in heavy lines, and the number of
complete tours is equal to the average of the back and front pitches.
Since the number of parallel circuits from brush to brush is q = 2,
the _E.M.F. equation of the wave-wound drum_ is E_a = pZ_a (N/60)[tau]
x 10^(-8) volts. Only two sets of brushes are necessary, but in order
to shorten the length of the commutator, other sets may also be added
at the point of highest and lowest potential up to as many in number
as there are poles. Thus the advantage of the wave-wound armature is
that for a given voltage and number of poles the number of active
wires is only 1/p of that in the lap-wound drum, each being of larger
cross-section in order to carry p times as much current; hence the
ratio of the room occupied by the insulation to the copper area is
less, and the available space is better utilized. A further advantage
is that the two circuits from brush to brush consist of elements
influenced by all the poles, so that if for any reason, such as
eccentricity of the armature within the bore of the pole-pieces, or
want of uniformity in the magnetic qualities of the poles, the flux of
each field is not equal to that of every other, the equality of the
voltage produced by the two halves of the winding is not affected
thereby.
In appearance the two classes of armatures, lap and wave, may be
distinguished in the barrel type of winding by the slope of the upper
layer of back end-connexions, and that of the front connexions at the
commutator end being parallel to one another in the latter, and
oppositely directed in the former.
[Illustration: FIG. 24.]
After completion of the winding, the end-connexions are firmly bound
down by bands of steel or phosphor bronze binding wire, so as to resist
the stress of centrifugal force. In the case of smooth-surface
armatures, such bands are also placed at intervals along the length of
the armature core, but in toothed armatures, although the coils are
often in small machines secured in the slots by similar bands of a
non-magnetic high-resistance wire, the use of hard-wood wedges driven
into notches at the sides of the slots becomes preferable, and in very
large machines indispensable. The external appearance of a typical
armature with lap-winding is shown in fig. 24.
The commutator.
A sound mechanical construction of the commutator is of vital importance
to the good working of the continuous-current dynamo. The narrow,
wedge-shaped sectors of hard-drawn copper, with their insulating strips
of thin mica, are built up into a cylinder, tightly clamped together,
and turned in the lathe; at each end a V-shaped groove is turned, and
into these are fitted rings of micanite of corresponding section (fig.
19); the whole is then slipped over a cast iron sleeve, and at either
end strong rings are forced into the V-shaped grooves under great
pressure and fixed by a number of closely-pitched tightening bolts. In
dynamos driven by steam-turbines in which the peripheral speed of the
commutator is very high, rings of steel are frequently shrunk on the
surface of the commutator at either end and at its centre. But in every
case the copper must be entirely insulated from the supporting body of
metal by the interposition of mica or micanite and the prevention of any
movement of the sectors under frequent and long-continued heating and
cooling calls for the greatest care in both the design and the
manufacture.
Forms of field-magnet.
On passing to the second fundamental part of the dynamo, namely, the
field-magnet, its functions may be briefly recalled as follows:--It has
to supply the magnetic flux; to provide for it an iron path as nearly
closed as possible upon the armature, save for the air-gaps which must
exist between the pole-system and the armature core, the one stationary
and the other rotating; and, lastly, it has to give the lines such
direction and intensity within the air-gaps that they may be cut by the
armature wires to the best advantage. Roughly corresponding to the three
functions above summarized are the three portions which are more or less
differentiated in the complete structure. These are: (1) the magnet
"cores" or "_limbs_," carrying the exciting coils whereby the inert iron
is converted into an electro-magnet; (2) the _yoke_, which joins the
limbs together and conducts the flux between them; and (3) the
_pole-pieces_, which face the armature and transmit the lines from the
limbs through the air-gap to the armature core, or vice versa.
[Illustration: FIG. 25.]
Of the countless shapes which the field-magnet may take, it may be
said, without much exaggeration, that almost all have been tried; yet
those which have proved economical and successful, and hence have met
with general adoption, may be classed under a comparatively small
number of types. For bipolar machines the _single horse-shoe_ (fig.
25), which is the lineal successor of the permanent magnet employed in
the first magneto-electric machines, was formerly very largely used.
It takes two principal forms, according as the pole-pieces and
armature are above or beneath the magnet limbs and yoke. The
"over-type" form is best suited to small belt-driven dynamos, while
the "under-type" is admirably adapted to be directly driven by the
steam-engine, the armature shaft being immediately coupled to the
crank-shaft of the engine. In the latter case the magnet must be
mounted on non-magnetic supports of gun-metal or zinc, so as to hold
it at some distance away from the iron bedplate which carries both
engine and dynamo; otherwise a large proportion of the flux which
passes through the magnet limbs would leak through the bedplate across
from pole to pole without passing through the armature core, and so
would not be cut by the armature wires.
[Illustration: FIG. 26.]
Next may be placed the "Manchester" field (fig. 26)--the type of a
divided magnetic circuit in which the flux forming one field or pole
is divided between two magnets. An exciting coil is placed on each
half of the double horse-shoe magnet, the pair being so wound that
consequent poles are formed above and below the armature. Each magnet
thus carries one-half of the total flux, the lines of the two halves
uniting to form a common field where they issue forth into or leave
the air-gaps. The pole-pieces may be lighter than in the single
horse-shoe type, and the field is much more symmetrical, whence it is
well suited to ring armatures of large diameter. Yet these advantages
are greatly discounted by the excessive magnetic leakage, and by the
increased weight of copper in the exciting coils. Even if the greater
percentage which the leakage lines bear to the useful flux is
neglected, and the cross sectional area of each magnet core is but
half that of the equivalent single horse-shoe, the weight of wire in
the double magnet for the same rise of temperature in the coils must
be some 40% more than in the single horse-shoe, and the rate at which
energy is expended in heating the coils will exceed that of the single
horse-shoe in the same proportion.
Thirdly comes the two-pole _ironclad_ type, so called from the
exciting coil being more or less encased by the iron yoke; this latter
is divided into two halves, which pass on either side of the armature.
Unless the yoke be kept well away from the polar edges and armature,
the leakage across the air into the yoke becomes considerable,
especially if only one exciting coil is used, as in fig. 27 A; it is
better, therefore, to divide the excitation between two coils, as in
fig. 27 B, when the field also becomes symmetrical.
From this form is easily derived the _multipolar_ type of fig. 28 or
fig. 29, which is by far the most usual for any number of poles from
four upwards; its leakage coefficient is but small, and it is
economical in weight both of iron and copper.
Materials of magnets.
As regards the materials of which magnets are made, generally speaking
there is little difference in the permeability of "wrought iron" or
"mild steel forgings" and good "cast steel"; typical (B, H) curves
connecting the magnetizing force required with different
flux-densities for these materials are given under ELECTROMAGNETISM.
On the other hand there is a marked inferiority in the case of "cast
iron," which for a flux-density of B = 8000 C.G.S. lines per sq. cm.
requires practically the same number of ampere-turns per centimetre
length as steel requires for B = 16,000. Whatever the material, if the
flux-density be pressed to a high value the ampere-turns are very
largely increased owing to its approaching saturation, and this
implies either a large amount of copper in the field coils or an undue
expenditure of electrical energy in their excitation. Hence there is a
limit imposed by practical considerations to the density at which the
magnet should be worked, and this limit may be placed at about B =
16,000 for wrought iron or steel, and at half this value for cast
iron. For a given flux, therefore, the cast iron magnet must have
twice the sectional area and be twice as heavy, although this
disadvantage is partly compensated by its greater cheapness. If,
however, cast iron be used for the portion of the magnetic circuit
which is covered with the exciting coils, the further disadvantage
must be added that the weight of copper on the field-magnet is much
increased, so that it is usual to employ forgings or cast steel for
the magnet cores on which the coils are wound. If weight is not a
disadvantage, a cast iron yoke may be combined with the wrought iron
or cast steel magnet cores. An absence of joints in the magnetic
circuit is only desirable from the point of view of economy of expense
in machining the component parts during manufacture; when the surfaces
which abut against each other are drawn firmly together by screws, the
want of homogeneity at the joint, which virtually amounts to the
presence of a very thin film of air, produces little or no effect on
the total reluctance by comparison with the very much longer air-gaps
surrounding the armature. In order to reduce the eddy-currents in the
pole-pieces, due to the use of toothed armatures with relatively wide
slots, the poles themselves must be laminated, or must have fixed to
them laminated pole-shoes, built up of thin strips of mild steel
riveted together (as shown in fig. 29).
[Illustration: FIG. 27.]
[Illustration: FIG. 28.]
However it be built up, the mechanical strength of the magnet system
must be carefully considered. Any two surfaces between which there
exists a field of density B_g experience a force tending to draw them
together proportional to the square of the density, and having a value
of B_g squared/(1.735 x 10^6) lb. per sq. in. of surface, over which the
density may be regarded as having the uniform value B_g. Hence, quite
apart from the torque with which the stationary part of the dynamo
tends to turn with the rotating part as soon as current is taken out
of the armature, there exists a force tending to make the pole-pieces
close on the armature as soon as the field is excited. Since both
armature and magnet must be capable of resisting this force, they
require to be rigidly held; although the one or the other must be
capable of rotation, there should otherwise be no possibility of one
part of the magnetic circuit shifting relatively to any other part. An
important conclusion may be drawn from this circumstance. If the
armature be placed exactly concentric within the bore of the poles,
and the two or more magnetic fields be symmetrical about a line
joining their centres, there is no tendency for the armature core to
be drawn in one direction more than in another; but if there is any
difference between the densities of the several fields, it will cause
an unbalanced stress on the armature and its shaft, under which it
will bend, and as this bending is continually reversed relatively to
the fibres of the shaft, they will eventually become weakened and give
way. Especially is this likely to take place in dynamos with short
air-gaps, wherein any difference in the lengths of the air-gaps
produces a much greater percentage difference in the flux-density than
in dynamos with long air-gaps. In toothed armatures with short
air-gaps the shaft must on this account be sufficiently strong to
withstand the stress without appreciable bending.
The magnetic circuit.
Reference has already been made to the importance in dynamo design of
the _predetermination of the flux_ due to a given number of ampere-turns
wound on the field-magnet, or, conversely, of the number of ampere-turns
which must be furnished by the exciting coils in order that a certain
flux corresponding to one field may flow through the armature core from
each pole. An equally important problem is the correct proportioning of
the field-magnet, so that the useful flux Z_a may be obtained with the
greatest economy in materials and exciting energy. The key to the two
problems is to be found in the concept of a magnetic circuit as
originated by H.A. Rowland and R.H.M. Bosanquet;[16] and the full
solution of both may be especially connected with the name of Dr J.
Hopkinson, from his practical application of the concept in his design
of the Edison-Hopkinson machine, and in his paper on "Dynamo-Electric
Machinery."[17] The publication of this paper in 1886 begins the second
era in the history of the dynamo; it at once raised its design from the
level of empirical rules-of-thumb to a science, and is thus worthy to be
ranked as the necessary supplement of the original discoveries of
Faraday. The process of predetermining the necessary ampere-turns is
described in a simple case under ELECTROMAGNETISM. In its extension to
the complete dynamo, it consists merely in the division of the magnetic
circuit into such portions as have the same sectional area and
permeability and carry approximately the same total flux; the difference
of magnetic potential that must exist between the ends of each section
of the magnet in order that the flux may pass through it is then
calculated _seriatim_ for the several portions into which the magnetic
circuit is divided, and the separate items are summed up into one
magnetomotive force that must be furnished by the exciting coils.
[Illustration: FIG. 29.]
The chief sections of the magnetic circuit are (1) the air-gaps, (2)
the armature core, and (3) the iron magnet.
The _air-gap_ of a dynamo with smooth-core armature is partly filled
with copper and partly with the cotton, mica, or other materials used
to insulate the core and wires; all these substances are, however,
sensibly non-magnetic, so that the whole interferric gap between the
iron of the pole-pieces and the iron of the armature may be treated as
an air-space, of which the permeability is constant for all values of
the flux density, and in the C.G.S. system is unity. Hence if l_g and
A_g be the length and area of the single air-gap in cm. and sq. cm.,
the reluctance of the double air-gap is 2l_g/A_g, and the difference
of magnetic potential required to pass Z_a lines over this reluctance
is Z_a.2l_g/A_g = B_g.2l_g; or, since one ampere-turn gives 1.257
C.G.S. units of magnetomotive force, the exciting power in
ampere-turns required over the two air-gaps is X_g = B_g.2l_g/1.257 =
0.8 B_g.2l_g. In the determination of the area A_g small allowance
must be made for the fringe of lines which extend beyond the actual
polar face. In the toothed armature with open slots, the lines are no
longer uniformly distributed over the air-gap area, but are graduated
into alternate bands of dense and weak induction corresponding to the
teeth and slots. Further, the lines curve round into the sides of the
teeth, so that their average length of path in the air and the air-gap
reluctance is not so easily calculated. Allowance must be made for
this by taking an increased length of air-gap = ml_g, where m is the
ratio _maximum density/mean density_, of which the value is chiefly
determined by the ratios of the width of tooth to width of slot and of
the width of slot to the air-gap between pole-face and surface of the
armature core.
The _armature core_ must be divided into the teeth and the core proper
below the teeth. Owing to the tapering section of the teeth, the
density rises towards their root, and when this reaches a high value,
such as 18,000 or more lines per sq. cm., the saturation of the iron
again forces an increasing proportion of the lines outwards into the
slot. A distinction must then be drawn between the "apparent"
induction which would hold if all the lines were concentrated in the
teeth, and the "real" induction. The area of the iron is obtained by
multiplying the number of teeth under the pole-face by their width and
by the net length of the iron core parallel to the axis of rotation.
The latter is the gross length of the armature less the space lost
through the insulating varnish or paper between the disks or through
the presence of ventilating ducts, which are introduced at intervals
along the length of the core. The former deduction averages about 7 to
10% of the gross length, while the latter, especially in large
multipolar machines, is an even more important item. Alter calculating
the density at different sections of the teeth, reference has now to
be made to a (B, H) or flux-density curve, from which may be found the
number of ampere-turns required per cm. length of path. This number
may be expressed as a function of the density in the teeth, and f(B_t)
be its average value over the length of a tooth, the ampere-turns of
excitation required over the teeth on either side of the core as the
lines of one field enter or leave the armature is X_t = f(B_t).2l_t,
where l_t is the length of a single tooth in cm.
In the core proper below the teeth the length of path continually
shortens as we pass from the middle of the pole towards the centre
line of symmetry. On the other hand, as the lines gradually accumulate
in the core, their density increases from zero midway under the poles
until it reaches a maximum on the line of symmetry. The two effects
partially counteract one another, and tend to equalize the difference
of magnetic potential required over the paths of varying lengths; but
since the reluctivity of the iron increases more rapidly than the
density of the lines, we may approximately take for the length of path
(l_a) the minimum peripheral distance between the edges of adjacent
pole-faces, and then assume the maximum value of the density of the
lines as holding throughout this entire path. In ring and drum
machines the flux issuing from one pole divides into two halves in the
armature core, so that the maximum density of lines in the armature is
B_a = Z_a/2ab, where a = the radial depth of the disks in centimetres
and b = the net length of iron core. The total exciting power required
between the pole-pieces is therefore, at no load, X_p = X_g + X_t +
X_a, where X_a = f(B_a).l_a; in order, however, to allow for the
effect of the armature current, which increases with the load, a
further term X_b, must be added.
[Illustration: Fig. 30.]
In the continuous-current dynamo it may be, and usually is, necessary
to move the brushes forward from the interpolar line of symmetry
through a small angle in the direction of rotation, in order to avoid
sparking between the brushes and the commutator (_vide infra_). When
the dynamo is giving current, the wires on either side of the diameter
of commutation form a current-sheet flowing along the surface of the
armature from end to end, and whatever the actual end-connexions of
the wires, the wires may be imagined to be joined together into a
system of loops such that the two sides of each loop are carrying
current in opposite directions. Thus a number of armature ampere-turns
are formed, and their effect on the entire system of magnet and
armature must be taken into account. So long as the diameter of
commutation coincides with the line of symmetry, the armature may be
regarded as a cylindrical electromagnet producing a flux of lines, as
shown in fig. 30. The direction of the self-induced flux in the
air-gaps is the same as that of the lines of the external field in one
quadrant on one side of DC, but opposed to it in the other quadrant on
the same side of DC; hence in the resultant field due to the combined
action of the field-magnet and armature ampere-turns, the flux is as
much strengthened over the one half of each polar face as it is
weakened over the other, and the total number of lines is unaffected,
although their distribution is altered. The armature ampere-turns are
then called _cross-turns_, since they produce a cross-field, which,
when combined with the symmetrical field, causes the leading
pole-corners ll to be weakened and the trailing pole-corners tt to be
strengthened, the neutral line of zero field being thus twisted
forwards in the direction of rotation. But when the brushes and
diameter of commutation are shifted forward, as shown in fig. 31, it
will be seen that a number of ampere-turns, forming a zone between the
lines Dn and mC, are in effect wound immediately on the magnetic
circuit proper, and this belt of ampere-turns is in direct opposition
to the ampere-turns of the field, as shown by the dotted and crossed
wires on the pole-pieces. The armature ampere-turns are then divisible
into the two bands, the _back-turns_, included within twice the angle
of lead [lambda], weakening the field, and the cross-turns, bounded by
the lines Dm, nC, again producing distortion of the weakened
symmetrical field. If, therefore, a certain flux is to be passed
through the armature core in opposition to the demagnetizing turns,
the difference of magnetic potential between the pole-faces must
include not only X_a, X_t, and X_g, but also an item X_b, in order to
balance the "back" ampere-turns of the armature. The amount by which
the brushes must be shifted forward increases with the armature
current, and in corresponding proportion the back ampere-turns are
also increased, their value being c[tau]2[lambda]/360 deg., where c = the
current carried by each of the [tau] active wires. Thus the term X_b,
takes into account the effect of the armature reaction on the total
flux; it varies as the armature current and angle of lead required to
avoid sparking are increased; and the reason for its introduction in
the fourth place (X_p = X_g + X_t + X_a + X_b), is that it increases
the magnetic difference of potential which must exist between the
poles of the dynamo, and to which the greater part of the leakage is
due. The leakage paths which are in parallel with the armature across
the poles must now be estimated, and so a new value be derived for the
flux at the commencement of the _iron-magnet_ path. If P = their joint
permeance, the leakage flux due to the difference of potential at the
poles is z_l = 1.257X_p x P, and this must be added to the useful flux
Z_a, or Z_p = Z_a + Z_l. There are also certain leakage paths in
parallel with the magnet cores, and upon the permeance of these a
varying number of ampere-turns is acting as we proceed along the
magnet coils; the magnet flux therefore increases by the addition of
leakage along the length of the limbs, and finally reaches a maximum
near the yoke. Either, then, the density in the magnet B_m = Z_m/A_m
will vary if the same sectional area be retained throughout, or the
sectional area of the magnet must itself be progressively increased.
In general, sufficient accuracy will be obtained by assuming a certain
number of additional leakage lines z_n as traversing the entire length
of magnet limbs and yoke (= l_m), so that the density in the magnet
has the uniform value B_m = (Z_p + z_n)/A_m. The leakage flux added on
actually within the length of the magnet core or z_n will be
approximately equal to half the total M.M.F. of the coils multiplied
by the permeance of the leakage paths around one coil. The
corresponding value of H can then be obtained from the (B, H) curve of
the material of which the magnet is composed, and the ampere-turns
thus determined must be added to X_p, or X = X_p + X_m, where X_m =
f(B_m)l_m. The final equation for the exciting power required on a
magnetic circuit as a whole will therefore take the form
X = A[Tau] = 0.8B_g.2l_g + f(B_t)2l_t + f(B_a)l_a + X_b + f(B_m)l_m. (3)
If the magnet cores are of wrought iron or cast steel, and the yoke is
of cast iron, the last term must be divided into two portions
corresponding to the different materials, i.e. into f(B_m)l_m +
f(B_y)l_y. In the ordinary multipolar machine with as many
magnet-coils as there are poles, each coil must furnish half the above
number of ampere-turns.
[Illustration: FIG. 31.]
Magnetic leakage.
Since no substance is impermeable to the passage of magnetic flux, the
only form of magnetic circuit free from leakage is one uniformly wound
with ampere-turns over its whole length. The reduction of the
_magnetic leakage_ to a minimum in any given type is therefore
primarily a question of distributing the winding as far as possible
uniformly upon the circuit, and as the winding must be more or less
concentrated into coils, it resolves itself into the necessity of
introducing as long air-paths as possible between any surfaces which
are at different magnetic potentials. No iron should be brought near
the machine which does not form part of the magnetic circuit proper,
and especially no iron should be brought near the poles, between which
the difference of magnetic potential practically reaches its maximum
value. In default of a machine of the same size or similar type on
which to experiment, the probable direction of the leakage flux must
be assumed from the drawing, and the air surrounding the machine must
be mapped out into areas, between which the permeances are calculated
as closely as possible by means of such approximate formulae as those
devised by Professor G. Forbes.
Excitation of field-magnet.
In the earliest "magneto-electric" machines permanent steel magnets,
either simple or compound, were employed, and for many years these
were retained in certain alternators, some of which are still in use
for arc lighting in lighthouses. But since the field they furnish is
very weak, a great advance was made when they were replaced by soft
iron electromagnets, which could be made to yield a much more intense
flux. As early as 1831 Faraday[18] experimented with electromagnets,
and after 1850 they gradually superseded the permanent magnet. When
the total ampere-turns required to excite the electromagnet have been
determined, it remains to decide how the excitation shall be obtained;
and, according to the method adopted, continuous-current machines may
be divided into four well-defined classes.
[Illustration: FIG. 32.]
The simplest method, and that which was first used, is _separate
excitation_ from some other source of direct current, which may be
either a primary or a secondary battery or another dynamo (fig. 32).
But since the armature yields a continuous current, it was early
suggested (by J. Brett in 1848 and F. Sinsteden in 1851) that this
current might be utilized to increase the flux; combinations of
permanent and electromagnets were therefore next employed, acting
either on the main armature or on separate armatures, until in 1867 Dr
Werner von Siemens and Sir C. Wheatstone almost simultaneously
discovered that the dynamo could be made _self-exciting_ through the
residual magnetism retained in the soft iron cores of the
electromagnet. The former proposed to take the whole of the current
round the magnet coils which were in series with the armature and
external circuit, while the latter proposed to utilize only a portion
derived by a shunt from the main circuit; we thus arrive at the second
and third classes, namely, _series_ and _shunt_ machines. The starting
of the process of excitation in either case is the same; when the
brushes are touching the commutator and the armature is rotated, the
small amount of flux left in the magnet is cut by the wires, and a
very small current begins to flow round the closed circuit; this
increases the flux, which in turn further increases the E.M.F. and
current, until, finally, the cumulative effect stops through the
increasing saturation of the iron cores. Fig. 33, illustrating the
_series_ machine, shows the winding of the exciting coils to be
composed of a few turns of thick wire. Since the current is undivided
throughout the whole circuit, the resistance of both the armature and
field-magnet winding must be low as compared with that of the external
circuit, if the useful power available at the terminals of the machine
is to form a large percentage of the total electrical power--in other
words, if the efficiency is to be high. Fig. 34 shows the third
method, in which the winding of the field-magnets is a _shunt_ or
fine-wire circuit of many turns applied to the terminals of the
machine; in this ease the resistance of the shunt must be high as
compared with that of the external circuit, in order that only a small
proportion of the total energy may be absorbed in the field.
[Illustration: FIG. 33.]
[Illustration: FIG. 34.]
Since the whole of the armature current passes round the field-magnet
of the series machine, any alteration in the resistance of the
external circuit will affect the excitation and also the voltage. A
curve connecting together corresponding values of external current and
terminal voltage for a given speed of rotation is known as the
_external-characteristic_ of the machine; in its main features it has
the same appearance as a curve of magnetic
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