THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA
VOLUME 24, NUMBER 6
NOVEMBER,
1952
Reiular Contributions
Duality in Mechanics
P. LE CORBEILLER AND ¾ING-WA ¾EUNG
CrufiLaboratory,
HarvardUniversity,
Cambridge,
Massachusetts
(ReceivedJuly 21, 1952)
Givenan electricnetworkE• whichhasM meshes
andP node-pairs,
its electricdualEl will haveP meshes
andM node-pairs
andits classical
mechanical
analogM! will haveM+ 1 nodes,M independent
node-pairs,
andP independent
nodecycles.A secondmechanical
systemM•, the classical
analogof El, will haveM
cyclesandP node-pairs.
If, for example,M= 2, P= 3, thesystems
E• andM•, analogs
in the Firestone
or
"mobility"method,will be governedby two meshequations,
expressing
that the algebraicsumof the
voltagesor velocitiesaroundany loopis zero;the systemsEi and Mj-• alsoFirestoneanalogs,will satisfy
two nodeequations,expressing
that the algebraicsumof the currentsor forcesleavingany nodeis zero.
Thesefour setsof equations
are identical,interchanging
symbolssuitably.The consideration
of the four
systems,
Ev,E6 My, Mf, forminga completeset,showsthe advantages
of theFirestone
overtheclassical
system
ofanalogies
andsuggests
a systematic
useof dualityin mechanical
aswellasin electrical
systems.
1. INTRODUCTION
one and the samemathematicalequation.Inductance
HE purpose
ofthispaperistopresent
a newway L in exampleEv is the analogof massM in example
of introducing
the "mobility"systemof electro- Mr, etc.
Considernext the same electric system E• and
mechanical
analogies
advocatedin 1933by Dr. F. A.
Firestone2
Alongtheway weshallhit uponan interest- another electric system E•, correspondingto each
of a current
ing methodof analyzinga mechanical
system,first otherby duality.HeresystemE• willconsist
i(t) in parallelwiththreebranches
C, G(= R-•)
suggested
by Dr. Firestone
•,2and recentlydeveloped generator
for thissecond
by Dr. H. M. Trent?Our ideais to bringtogethertwo and I'(=L -•) (Fig. 3.). The justification
type
of
correspondence
is
that
the
differential
equations
types of transformation
much employedin system
analysisand synthesis,
electromechanical
analogyand for the two systemsare identical,with only a change
duality,andto makecomplete
useof theirpossibilities.in symbols(L-C, i-v, etc.):
We shall assume that the reader is familiar with the
"classical"
systemof analogy(force--emf,velocity-current)andwithdualityin electricnetworks
(exchange
of currentand voltage,exchangeof seriesand parallel
connections),but not with Dr. Firestone's system.
The symbolsusedwill befor the mostpart thoserecommendedjointly by the ASME and the AIEE.
2. THE
FUNDAMENTAL
FOUR-SYSTEMS
SET
Ld•+
Ri+
Sf idt=v(t),O)
C--+Gv+I'
dt
f vdt=i(t).
(3)
The two electricsystemsEv, E• and the mechanical
systemMf arenow threedistinctexamples
of realization of one and the samemathematicalequation.We
theirsymbols
in thefollowing
way:
Consideran electricsystemE• and the mechanical canarrange
systemMi which is its classicalanalog.As simple
examples
wechoose
a series
circuit(L, R, S) containing
a voltagegeneratorv(t) (Fig. 1), and its mechanical
analog(M, D, K), acteduponby a forcef(t) (Fig. 2).
E•
The justificationfor this correspondence
is that the
differentialequationsfor the two systemsare identical, The horizontaldoublearrow is for "classicalanalogy":
the vertical doublearrow is for "duality." Obviously
with onlya changein symbols(L-M, i-v, etc.):
a fourthsystemis missing(a fourthrealizationof the
di
L--+ Ri-b Sq= v(t),
(1) sameequation!).This mechanicalsystemM• should
dt
be the classical
analogof E•, and shouldcorrespond
to
M/by (mechanical)
duality.We don'tknowyet what
dv
M--+ Dv+ Kx= f(t).
(2) this systemis, but it is easy to write its differential
dt
equation,derivingit from Eq. (2) in the sameway
The electricsystemE• and the mechanicalsystem Eq. (3) canbe derivedfromEq. (1). We obtain
Mi are thus two distinct examplesof realizationof
• F. A. Firestone,J. Acoust.Soc.Am. 4, 249 (1933).
• F. A. Firestone,J. Appl. Phys.9, 373 (1938).
K dt
aH. M. Trent, "An Alternative Formulation of the Laws of
Mechanics"(Naval ResearchLab., September,1950); Am. Soc.
Mech. Eng. PaperNo. 51-A30,(November,1951).
643
D
M
SystemM• must then consistof threemechanical
elementsK, D, M in series,through which the same
1544
•
P.
L
?
LE
CORBEILLER
R
Fro. 1. A series elec-
tric circuit •.•.
AND
¾.
YEUNG
ent node-pairs.
In this paperwe confineourselves
to
the simplecasewhereeverymobilenodein the system
isdefinedin spaceby a singlecoordinate,
for themoment
a translationalcoordinatex. The configurationof the
systemwill then be definedat any time by the values
of P coordinates,
and the P equationsof the system
will expressthe Newton-d'Alembertlaw that at every
mobile node the algebraicsum of the applied forces
andof the inertialforce(if present)mustbe zero.These
forcefit) will be acting;the sumof the threerelative P equations will determine the unknown velocities of
velocities
df/Kdt,fiD, ,ffdt/M, mustequalthevelocity the mobilenodeswith regardto ground.
v(t) imposedat somepoint of the system.The system The analysisof systemM• (Fig.4) bringsa surprise.
shownon Fig. 4 satisfies
theseconditions.
It containsa Every term in Eq. (4) is a velocity.The equation
massM, a dashpotof viscousresistance
D and a spring expressesthat the sum of the relative velocities of
of stiffnessK in series,and a crank-and-rodvelocity nodeA with regard to nodeB (acrossthe spring),of
generator,alsoin series.(In all our drawingswe shall B to C (acrossthedashpot),andof C (the mass)to the
assumethat no solidfriction is presentanywhere,and wall gg (whichis here the reference
systemand thus
a viscous
resistance
will alwaysbeexplicitlyrepresented servesthe role of "ground")is equal to the velocity
by a dashpot.)
imposedto nodeA with regardto gg.Thus,if weconsider
The set of four systemsE• E•, Ms, M, is obviously the closedloop(gABCg),Eq. (4) expresses
the correct
a complete
set(like two blue and red trianglesand two ff unfamiliarstatementthat the algebraicsum of the
blue and red circles).Until we had this completeset relativevelocities
of everynodein the loopwith regard
beforeoureyeswe couldnot havea goodunderstanding
to the oneon the right mustbe zero2.•
of the possibilities
of the two transformations,
analogy
The generalizationof this is obviouson the basisof
and duality.
what we knowof duality andanalogy.In suchmechan3. A NEW
ANALYSIS
OF
MECHANICAL
SYSTEMS
Equations(1) and (3) for the electricalsystemsare
the simplestexamplesof Kirchhoff'sLaws. Kirchhoff's
voltagelaw statesthat the algebraicsumof the voltages
around any closed circuit or mesh must be zero.
SystemE, hasonly onemeshandoneunknowncurrent
ical systemsas will be consideredin this paper there
will be a certain number M of independentloops,and
theirequations(Zv= 0 aroundeachloop)will determine
M unknown forces, F•, F•,...
Fu, the force F}
acting through all the elementsconnectedin series
aroundthe kth loop.
It
is well known
that
an electric
network
can be
i(t) (Fig. 1). In generalthere will be in an electric analyzedeitheron a meshbasis(21v=0,meshcurrents
network a certain numberM of independentmeshes, unknown)or on a node-pairbasis(2;i=0, voltages
and their equationswill determinethe M unknown across
node-pairs
unknown).In mostcasesoneof these
mesh currents.
methodswill lead to fewer equationsthan the other.
Kirchhoff'scurrent law states that the algebraic It appears that a mechanicalsystem of the type
sum of the currentsleaving any node must be zero.
SystemE• has two nodes,A and the groundg (Fig. 3)
(pointsat the samepotentialconstitutea singlenode).
NodesA and g providethe sameequation(3); the only
unknownis the voltage acrossthe •ode-pair Ag. In
generalthere will be in an electricnetwork a certain
number P of independentnode-pairs,and their equations will determinethe unknownvoltagesacrossthe
P node-pairs.
consideredhere can be analyzedeither on a node-pair
basis(2;f=0, relativevelocitiesunknown),or on a loop
basis(Zv= 0, forcesunknown).We may call for short
the Newtori-d'Alembertequations2:f= 0 the dynamical
or the force equations,and the new equations2•v=0
the k/nemat/cor the velocityequationsof the system.
This new, circuital formulation of the laws of
mechanics,in contrastwith the classical,nodalformula-
tion, has been emphasizedin a recentpaper by Dr.
Considernow the mechanicalsystemMs (Fig. 2).
We may say that all the pointsin a mechanicalsystem
whichare rigidly connectedto eachother constitutea
single node. This system then has two nodes, the
massless
horizontalconnecting
rod AA and the ground
gg. Equation (2) expressesthat the algebraicsum of
the forcesapplied to node A is zero. From the standpoint of classicaldynamics(--Dr), (--Kx), and f(t)
are appliedforces,while --M(dv/dt) is an inertial or
//////lift/Ill/1/Ill//////////////////////IX
d'Alembertforce.In generalthere will be in a mechanFro. 2. A mechanicalsystemM!, the "classical"analogof the
ical systema certain number P of mobilenodes,in electric
circuitE, in Fig. 1. AA is a rigid massless
rod constrained
additionto the "fixed" groundg, and thusP independ- to remainhorizontal.D is a dashpot.
DUALITY
IN
MECHANICS
645
H. M. Trent, usinga differentapproachfrom that of
the presentpaper."
4. A PHILOSOPHICAL
ASIDE
The readermay wonderhow it is possibleto derive
the dynamicalbehaviorof a systemfrom kinematic Fro. 4. A mechanicalsystemMo, the classicalanalogof the
equations.This questionarisesbecansewe are accus- electric circuit E/ in Fig. 3. The dashpot casingand the mass
tomedin dynamicsto a mistakenemphasison forces, glide without friction on the plane gg'.
traceableto a metaphysicalattitude current in Newton's times.From this emphasisoa forceconsidered which make up the completeset can be described
as the "cause"of motion,therefollowedin electricity equallywell by nodeequationsor by meshequations,
are identicalfor the four systems
an emphasison electromotiveforce consideredas the and theseequations
"cause" of the current flow. To one without such
if we disregardthe changesin symbols.
Let usassume,
thattheelectric
network
Evhastwomeshes
preconceptions,
it shouldbe clear that in Ohm's law fore.xample,
E=IR, as well as in Newton'slaw f=ma, the general and three node-pairs.Then it will be most simply
by twomeshequations,
Ev=0 (meshcurrents
gas law PV=nRT, or any natural law whatsoever, described
to Eoby duality,
thereis no causeand no effect,there are only variable unknown).ThenEi, whichcorresponds
physical quantifies permanently connected by a will be most simply describedby the sametwo equamathematicalequation. For instance,wheneverany tions,whichare now nodeequations,Y.i=0 (voltages
two of the quantitiesE, I, R are knownat any time t, acrossnode-pairsunknown).Then M/, the classical
Ohm's law allowsus to find the third one; if I and R analogof E•, .will be most simplydescribedby the
whicharehereZf= 0 foreverynode
happento be known,it wouldmakelittle senseto call sametwoequations,
them the "cause"of E, and call E their "effect." The (relative velocitiesunknown);and M,, the dual of
Mr, will be most simplydescribed
by the sametwo
equations,which are here loop equations,
(forcesunknown).(Let the readercheckthis on the
electric circuit Ei, correspondingby duality
systems
of Fig.6, andonthecorresponding
equations.)
G
F
•".-,
to the series circuit E•
inFig.
1.
Of course,the four systems
arebasicallyequivalent;
they are just four examples
of realizationof the same
set of equations.However,if on a lessabstractplane
I__[ we ask ourselveswhich of the two mechanicalsystems
M/, M,, most "resembles"the electricalsystemF_•,
it is clear that M, is the unequivocal
answer,sinceE,
sameremarkappliesthroughoutthe field of classical and M• both have two meshesand three node-pairs
physics.
and are described
by the sametwo meshequations.
In the analysisof electriccircuitswe make use of
SimilarlyEi and Mf are mostalike,sincethey both
three suchlaws or relations,which are (usingp for havetwonode-pairs
andthreemeshes
andaredescribed
shorteitherford/dror forj•oin thealternating
steady- by the sametwo node-pairequations.The classical
state)
analogymakesthe meshesof E, correspond
to the
•= q/C= i/Cp, v= Ri, v= Lpi.
(5) node-pairs
of M•, an awkwardsituationdue to the
Similarlyin the analysisof mechanical
systemswe topologicalinnocenceof our forebears.The only
analogysuited to this enlightened
make use of three relations,relative to a spring,a electromechanical
age
is
that
between
EoandM•, andbetween
E• andM•;
dashpot,and a mass:
and that is the one axtvocated
by Dr. Firestone(who
f=Kx=Kv/p, f=Do, f=Ma=Mpv.
(6) callsit the "mobility"system,references
1, 2).
We,
therefore,
adopt
this
analogy,
and
in
consequence
Each one of these relations is characteristic of a
presentthe arrayof our four systems
or examples
in
singlemechanical
element,andrelatestheforcetkrougk the new and better form:
the elementto the velocity acrossit. It is basically
indifferent(whenK, D, and M are given)whetherwe
solvethemfor the forceor for the velocityand combine
thesein the dynamical or in the kinematic equations.
Forces and velocities will be connected in either case.
5. THE •MOBILITY"
ELECTROMECHANICAL
SYSTEM
OF
ANALOGIES
A remark forcesitself upon us at this stageof the
argument.Any oneof the four systems
E•, Ei, M.t, M,
646
P.
•
K
0
LE
CORBEILLER
AND
Y.
YEUNG
and equationsof one of the examplesof the complete
set.There may be constractional
reasonswhy in a given
caseM/should be preferred to M, or vice-versa,and
M
the samefor Ei, E,. The displayof the completeset
.•///////////////////////////%
enables one to choose the most convenient
ONE MESH : õASCg
ONE LOOP: gASCg
I f
I
t
(1•)
ONE NODE A, PLUS GROUND
ONENODEA, PLUS 6ROUN[•
We shall immediately follow our own advice and
bring togetherthe examplesof Figs. 1 to 4 (Fig. 5).
We,arenowusingthe samegraphicalsymbolsfor alternatingconstant-velocity
and constant-force
generators
asfor constant-voltage
andconstant-current
generators,
a circleand an oblong,respectively,with two terminals
each; this makes mechanical diagrams very close
copiesof their electricalanalogs.The samesymbolv is
usedfor velocityand voltage,whichhappensto fit the
Firestone"mobility" systemof analogy.
I-'
K
-I*Mpv +Ov•--•-v : 0
-•. +Cpv+Ov• •-v = 0
FIO. 5. Four systems•two mechanical,M• Mr, two electrical,
E,, El, forming a completeset. Two systemsin the same row
correspondby the Firestonesystem of analogy; two systemsin
the samecolumncorrespond
by duality.
The horizontal
realization.
double arrows are for electromechan-
6. CORRESPONDENCE
BETWEEN
ELECTRICAL
MECHANICAL
AND
QUANTITIES
There are nowfour setsof mechanicaland electrical
quantitiesto be considered,
whichcorrespond
two by
two either horizontally (by "analogy") or vertically
(by duality).We accordingly
divideour paperinto
ical analogy (Firestone,of course);the vertical ones
for duality, either between electrical or between mechanicalsystems,a topologicaltransformationexchangingnode-pairsand meshes.
The orderof rowsor columns
in the array is indifferent. Dotted diagonalsindicate
"classical"analogy.It is clear that classicalanalogy
is doing two thingsat a time: exchangingnode-pairs
and meshesand exchangingmechanicaland electrical
quantities (going back to the trianglesand circles
(b)
quotedin Sec.2, it makesa blue trianglethe analogof
a red circle).
FIG. 7. (a) An ideal 1:1 transformer,
closedon a capacitorwith
oneplate grounded;(b) its mechanical
analog.
We submitthat any oneinterested,inthe analysisor
synthesisof electricaland mechanicalsystemsshould
alwaysas a matter of routine divide his sheetof paper four quarters.In thoseon the left we put mechanical
into four quartersandput downin eachonethe diagram quantities,in thoseon the right electricalquantities.
When two mechanicalelementscorrespondto each
other by duality we place them symmetrically
with
regardto the horizontal(mirror) line. We do the same
ß
J.
for electricalquantities(see,for instance,the pair
velocity-force,
the pair capacity-inductance,
etc.). We
thus obtain the following completecorrespondence
,•
TWO LOOPS;f• ,f• UNKNOWN
TWOMESHES;t• ,i2UNKNOWN
(•) V •11
tI+C;IP
(2) •pp
(ia-'•,}
'(•-•+•-),,- 0
f
scheme:
We note that coordinateand momentumare dual,
as alsotheir analogsmagneticflux and electriccharge.
Dualityconsists
essentially,
ratherthanin theexchange
of, and i (or v andf) as in elementarypresentations,
in the exchangeof the coordinateand momentum of
eachparticle.It is actuallythe simplestexampleof a
transformation
whichis of importancein analytical
dynamics
4 and in modemphysics.
TWONOOE$*G•D,;
vA,',,•
UNKNOWN
(A}-•,*C•pv,•*.•-(v,•-ve}
=0
Fro. 6. Four examplesof a one-sectionlow-passfilter.
(Arrowsf•, f2 shouldpoint downwards.)
7. LOW-PASS
FILTER
We shall give now the four "examples"of a onesection
low-pass
filter,a simplesystemdefinedby two
4H. Goldstein,
Classical
Mechanics
(Addison-Wesley
Press,Inc.,
Cambridge,1950),p. 245.
DUALITY
IN
MECHANICS
647
TAnLE I. Correspondence
schemebetween mechanicaland
electricalquantities,and between"duals."
velocity
v
force fM
Imass
[ stiffness
K
{voltage
l'capacityC
f momentum
p= My
] electric
charge • q= Cv
viscositycoeff.
-%
current
D
conductance
G
[ recipr.inductanceP: L-a
TWOLOOPS,q• ,q2UNKNOWN
[coordinate
I x=K-if [magnetic
flux [4,=
Li
momentum
•
charge
.electric
magnetic
flux
(gBg)
d•-(•2-%,)* •'g2
I
•0
__e iiiiiln:
[ reclpr.
mass
M-a
susceptance S= C-1
I recipr.risc. coeff. D-I
[ compliance
K -a
j velocity
• force
R
L
e
v
f
v
TWONODES
• ORO;(.g,.L,OB
UNKNOWNTWONODES"
GRD;v,,VeUNKNOWN
(A)
-q,.JpCOa
+.-• t(.,3•L.O6)
:0
(s)•2(c%-%)
+o'c%:
o
is)•-(%-%)
+0%, o
equations,as further application of the preceding
remarks(Fig. 6). It will be seenthat all four setsof
two equationsare identical.
It is interestingto observethat by addingthe force
equations(A) and (B) of systemM/we obtain
FIG.
9. Four systems,two rotational,M,o, Me, t•vo electrical,
E•, E6 forminga completeset.
the symbolv represents
the relativevelocityof the two
ends at time t (with suitablesign conventions);its
Thiscanbeconsidered
asthe equationZf= 0 relativeto value is thereforeindependentof the referencesystem.
"ground." It thus appearsthat in establishingthe P A point-mass
or particle,however,constitutes
a single
force equationsof a mechanicalsystem,the ground node,and the "fundamentalequationof dynamics,"
canbe substituted
for anyoneof the P othernodesin
f= Ma = Mpv= Mp•x,
(6")
the system,as we know to be the casein electricity
(communication
of Dr. Trent).
holdsonly whenx is the coordinate
of that particleor
-fq-M•pvA+ M2pvB+D•-- O.
nodein an "inertial" referencesystem.It would not
be true, for instance,if x representedthe distanceof
The point-masses
in a dynamicalsystemrequirea the particle in questionto someother particle, itself
different treatment from that of the other five basic
in oscillatorymotion with regard to the laboratory
mechanidal and electrical elements.
flooror walls (which are usuallya satisfactoryinertial
A springand a dashpothave two ends (nodes, referencesystem).In orderto remindourselves
of this
terminals)and in the two equations.
very importantpoint,we shallfollowa suggestion
of
Dr. Trent and draw an interrupted line from each
f= Kv/p, f= Dr,
(6')
mass-point
or particleto ground,in the directionof
8. SPECIAL
CASE
OF A MASS
its coordinate X.
It followsthat in the strictanalogof a givenmechan-
ical system,every capacitorshouldhave one of its
platesconnected
to ground.The twoequations.
f=Mpv,
7o,/;//•////?///)///////?/,•o•
ONE NODE-PAIR
v UNKNOWN
A If• A
ONE
NODE-PAIR.
Mi
•
ONE LOOP (gABg)
v UNKNOWN
I UNKNOWN
M•p+D, M2p+ p
t2•
•'C2
ONE MESH
* UNKNOWN
c,p,o,
Fro. 8. Four systemsforminga completeset, oneof whichrequires
the adjunction of an ideal 1:1 transformer.
i=Cpv
(7)
will then representequivalent physical situations,,
and v representingthe velocity of a particle and the
voltageof a specificcapacitorplate, both with regard
to a groundpermanentlyat rest.
On a given electric diagram it may happen that
neitherplate of a capacitorcanbe grounded.If however
it is necessary
to do so, onecan replaceeachcapacitor
by one windingof an ideal 1: ! transformer,
closethe
other winding on the capacitor,and ground one of
theplates(Fig.7(a)).Thiswillbringnochange
in the
equationsof the system. It is then the arrangement,
the mechanicalanalogof whichwe want to find.
The mechanicalarrangementon Fig. 7(b) answers
the question.It consistsof three massless,
rigid bars,
648
P.
LE
CORBEILLER
la3, 264,and ab.Bars13 and 24 havethe samelength,
and a, b are their middlepoints.All hingesare frictionless.Node 4 or g is fixed with regard to the groundor
wails; node3 is the massM to whichforcef is to be
applied.Nodes1, 2 can have independentbut small
vertical displacements,
hencebar ab remainsvertical.
The forcesactingon all four.nodesare equal,and the
velocityof 2 with regardto 1 is equalto the (absolute)
velocityof 3 with regardto g, choosing
positivedirections as indicated. 5.6
Figure8 showsfour "examples"of a simplesystem,
one of which contains a mechanical 1:1 transformer.
AND
Y.
YEUNG
directionin a loopwith regardto the oneon the right.
Theseare the M kinematicequationsof the system.
There will be in the rotational system as many
independentmobile nodes as there are independent
node-pairs in the analog electric network. Several
torquesq• may be acting on the samenode, as well as
an inertialor d'Alemberttorque(--JO). The analogof
Kirchhoff'scurrent laws is that 2q must be zero for
everymobilenode.Theseare theP dynamicalequations
of the system.
The equationsin Fig. 9 will make thesecondensed
explanations
clear.
It shouldbe noticedthat this difficultyis not special
to the Firestonetype of mechanical-electrical
analogy.
In the classicalanalogyit is not possibleto draw the
systemMo, corresponding
to El, withoutmakinguseof
a 1:1 mechanicaltransformer.However, the difficulty
is harder to analyze than in the Firestonesystem
becauseof the complicationresultingfrom the interchangeof node-pairsand meshes.
lO. CONCLUSION
The above presentationof electromechanical
analogiesemphasizes
the role of duality in the analysisof
mechanical
as well as of electricalsystems.This point
of view enablesone to carry over to a broadclassof
mechanicalsystemsthe resultsof the researchdoneon
the meshand nodeequationsof electricnetworksby a
numberof workers,notablyby Mr. GabrielKron.?
9. MECHANICAL
ROTATIONAL
SYSTEMS
The Firestoneor "mobility"systemof analogies
is
in the followingway: First it is shown
To avoid lengthyrepetitionof most of the above usuallypresented
material, the treatment of rotational systemsw/ll be to be a "possible"system,just as workableas the
explained on the four examplesof a single system classicalone; then one remarks that electrical circuits
(Fig. 9). Particlesfree to move on a line are now derivedon the mobilitybasisresembletheir mechanical
replacedby particlesfree to rotate around an axis, analogsvery closely,whichis obviouslyan advantage.
that is, free to move on a c/rcumferencecentered on No explanationis given of why it shouldbe so, nor do
we knowwhethera third type of analogymight not be
the axis.Sucha particleA is definedby the angleof its
possibleand perhapsbe preferableto the two others.
radiusvectorOA with a fixeddirectionOg whichserves
Once we realize that mechanicalsystemscan be
as referencesystemor "ground,"and by a momentof analyzed,just like electricnetworks,on either a mesh
inertiaJA (whichreplaces
the mass).Severalparticles, or a node-pair
basis,wecanapplythenotion.ofduality
rigidly connectedto a singleshaftor sleeve,constitute to mechanical
systems,
and it followsthat two electric
a single
node.
(Thus,a whole
diskorflywheel
isonenode, andtwo mechanical
systems,
dualtwoby two,formin
with a specificmomentof inertia J; the coordinateof a sensea completeset. The Firestoneelectromechanical
the nodeis the variableangleof any radiusOB or OA, analogyis then most naturallydefinedas the one in
which an electric node correspondsto a mechanical
paintedon the flywheel,with the fixeddirectionOg.)
One can form with the distinct mobile d[rections
node, an electric mesh to a mechanicalloop. The
OA, OB . .., and the "ground"Og,asmanyindepend- considerationof a completeset of four systemsis
ent loops (gAB...
g) as there are independent believedto be originalwith the presentpaper.
It shouldbe noted that we have only covereda
meshesin the analogelectricnetwork. The analogof
Kirchhoff'svoltagelawsis that Zo•mustbe zeroaround certainclassof mechanicalsystems,that in whichthe
of everyparticleis definedby one coorevery loop, whereo0is the angular velocity of every displacement
dinate only. If mechanicalparticleswith two or three
• H. F. Olson,DynamicalAnalogies(D. Van Nostrand Company,
Inc., New York, 1943).
6 H. M. Trent, PrtwtlealAspectsofE lectromechanlcal
Analogies,
N.R.L. EngineeringColloquinm,June 26, 1947 (unpublished
memorandum).
coordinatesare considered,entirely new possibilities
appear,as the exampleof a gyro plainly shows.
?G. Kron, TensorAnalysisofNetworks
(JohnWiley& Sons,Inc.,
New York, 1939).