A NEW
ANALOGY
BETWEEN
ELECTRICAL
MECHANICAL
AND
SYSTEMS
:ByF. A. Frv•s•o•
Universityof Michigan
By considering
eachmassin a linear mechanicalsystemashaving two terminals,
onefixedin the massandonefixedto theframeof reference,
everylinearmechanical
systemis reducedto a multiplicityof closedmechanicalcircuitsto whichforce and
velocityrelationssimilar to Kirchhoff'slaws, may be applied. The conventional
mechanical-electrical
analogyis derivedfrom the similarityof the equationsv--f/z
andl--E/Z. It is incomplete
in the followingrespects
whichlead to difficultyin its
application.
which
I. There
indicates
is aa lack
fundamental
of analogy
difference
in the use
in the
of nature
the words
of the
Uthrough"
analogous
and
quantities•
•across"
for instance,forcethroughand e.m.L across.
II. Mechanicalelementsin seriesmustbe represented
by electricalelementsin
parallel,and viceversa.
III. Mechanicalimpedances
in seriesmustbe combinedas the reciprocalof the
sum of the reciprocalswhile electricalimpedancesin seriesare additive.
IV. There is an incompleteness
in the mechanicalanaloguesof Kirchhoff'slaws.
The new analogyis derivedfrom the similarity of the followingequations:
v--f• andE-- IZ where• is the reciprocalof the mechanicalimpedance
asusuallydefreed.This newanalogyis completein all of the above-mentioned
respectsin which
the old analogyfailed.It leadsto analogous
relationsof a simplesortand permitsan
equivalent
electrical
circuitto be drawnin an easyir•tuitivemanner.
INTRODUCTION
If a groupof physicalconceptsor quantitiesare related to eachother
in a certain manner, as by equationsof a certain form, and another
group of conceptsor quantities are interrelated in a similar manner,
then an analogymay be said to existbetweenthe conceptsof the one
groupand thoseof the othergroup.The essentialelementoœan analogy
is a similaritybetweenthe relationswithin onegroupand the relations
within the other group.Two groupsof conceptsmay be analogousin
somerespectsand not analogous
in otherrespects.
An analogyis valuableto the extent that it permitsa knowledgeoœ
one field to be applied in another field. Sincemuch more is known of
the characteristics of electrical circuits than of certain kinds of mechani-
cal systems,it is often valuableto discussa mechanicalsystemin terms
of its electricalanalogue.It will be shownthat the conventionalmechanical-electrical
analogy is incompletein certain important par-
ticularswhichmake it difficultto apply in practice.A new kind of
mechanical-electrical
analogyis setforth belowwhichis morecomplete
than the old and permitsan equivalentelectricalcircuit to be drawnin
249
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
250
JOUP.
NALO1•THEACOUSTICAL
SOCIETY
[January,
a muchmorestraightforwardand common-sense
manner,not requiring
suchcarefulreasoningat eachstep.
We have all learned at school that inductance
in an electrical circuit
playsa part similar to massin a mechanicalsystem.• In substantiation
of thisviewpointit is mentionedthat the energystoredin the magnetic
field of the, inductanceis « LI •' and the kinetic energy of a massis
« m•; that the inductancetends to prevent a changeof current by
generating
a backe.m.f.of magnitudeLdI/dT just asa masstendsto
prevent a changeof velocity by producinga reactingforce of magnitude md•/dt. But onemight say with equaltruth that capacityplays
the r51eof electricalmassbecausethe energystoredin the electrostatic
fieldof a condenser,« CE•, corresponds
to the kinetic energyof a mass,
• m•; and alsothat the condensertendsto prevent a changeof e.m.f.
by absorbinga currentof magnitudeCdF./dtjust as a masstendsto
prevent a changeof velocity by producinga reacting force of magnitudemd•/di.
Similarly we have been told that capacity in an electricalcircuit
plays a part similar to the compliancec of a spring in a mechanical
system.And it is pointedout that the energystoredin the electrostatic
field of the condenseris « CE• corresponding
to the energy« cf•'stored
in a springby a forcef; that a condenserwill hold a charge,CE, proportionalto the e.m.f. while a springundergoesa displacement,cf, proportionalto the appliedforce.But it may be saidwith equalreasonthat
inductanceplays the part of electrical stiffnessbecausethe energy
stored in the magnetic field of the inductance, « LI •, correspondsto
the energy stored by a spring,« cf•; and also that an inductancewill
storea voltageimpulse(fEdt) of magnitudeLI proportionalto the
currentwhile a springundergoes
a displacement,cf, proportionalto the
appliedforce?
It is, therefore,evidentthat a new analogyis possiblein which force
is identifiedwith current rather than with e.m.f. as in the old analogy.
In what follows,it will be shownthat the new analogy is more complete and moreusefulthan the old.
THE MECHANICAL
ELEMENTS
We will confine our attention to the consideration of those mechanical
systemswhereinall the forcesand the resultingvelocitiesare essentially
• For instance,Starling,ElectricityandMagnetisdn,
p. 307.
a In Liven's, TtteoryofElectricity,p. 416, it is said uIt must,however,be particularlyemphasizedthat we have no definiteproofthat this (magnetic)energyis kinetic,it is merelya
matter of convenientchoiceso to regardit. •
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. FIRESTONE
251
in oneline. Many vibrating systems,includingtorsionalsystems,can
be very simply reducedto sucha linear problem.The elementsof a
mechanical
systemareassumed
to be of threekinds,springs,resistances
andmasses,althoughpracticallyit may be difficultto realizeany oneof
themin its pureform. We may assumethat all of the elementslie in one
horizontalline and that the positive direction along this line is from
left to right.In orderto understand
theanalogyin detail,it isnecessary
to look carefully at the nature of eachkind of mechanicalelement and
to be surethat we havea clearconceptionof the meaningof the quantities whichmay be usedin describingits state.
A springhas two terminalsto which forcemay be applied,and since
the forceis the samein all parts of the springwe will speakof the
"forcethroughthe spring."The forcethroughthe springmay be assumedto be positivewhenthe springis undertension,and the fact that
a springis undertensionmay be indicatedby an arrowin the positive
-•v
-
Fro. 1. Spring.
-I-¾
--
Fro. 2. Re,sistanc•.
Fxo. 3. Mass.
direction(to the right) asshownin Fig. 1. Thisis merelya convention
which must be remembered.There will always be a positive force
througha springwhenit is longerthan its unstrainedlength. •If the left
terminalof a springthroughwhichthereis a positiveforce(tension)is
attachedto someobject,then the springlieson the positivesideof the
objectand the objectwill be subjectedto a forcein the positivedirection. If the right terminalof the springis attachedto the object,then
the springlieson the negativesideof the object,and the positiveforce
throughthe springwill pull the objectin the negativedirection.
The relative velocity of the terminalswill be calledthe "velocity
difference
across"the springor simplythe "velocityacross."The velocitydifference
is countedpositiveat a giventerminalif in imagining
that terminal as fixed, the other terminal is movingin the positive
direction.This may be indicatedby + and - signsasin Fig. 1. Thus
if westandat the left endof the springandimagineit asfixed,the plus
signby our side will indicate that the other terminal is moving in the
positivedirection(to the rightin the figure)or awayfromus.Similarly,
the negativeat the right end indicatesthat if the right end is fixed,
theleftendwillbemovingin thenegative
direction
(to theleft). Conse-
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
252
JOURNAL
O1'THEACOUSTICAL
SOCXET¾
[January,
quently, a plus sign at the left terminal of a springindicatesthat the
springis growinglonger,and viceversa.(In a torsionalspringthereis
no essentialdistinction between the two possibledirectionsof twist
analogousto lengtheningand shorteningin a linear spring. One may
merely assume arbitrarily that a counterclockwisevelocity of the
nearer terminal relative to the farther is a positive angular velocity
difference across the spring; and likewise that a counterclockwise
torque on the nearer terminal sends a positive torque through the
spring.)
A mechanicalresistancemay be visualizedas two massless
concentric
tubes with a layer of viscousoil between. It also has two terminals.
With the sameconventionsas in the previousparagraphand as indi-
catedin Fig. 2, a resistance
will have a positiveforcethroughit (tension)when the velocitydifferenceacrossit is positiveat the left terminal (resistance
growinglonger).
It is not obviousthat a masshas two terminals.One cannotapply a
forceto a springor to a resistance
without graspingit at two points,but
forcecan appa•ntly be appliedto a massby contactingit at onepoint
only. But a force is itself a two-terminal element sinceaction and reaction are ,qual and opposite;it is impossibleto pushor pull without
standingon something.Thus in order that a forcemay act on a mass,
the force must react againstanother mass,which may be the earth.
The accelerationproducedon a mass by a given force is the same
whether that force reactsagainst a small massor againstthe earth.
Sincewe measurethe velocityof the massrelative to the earth,• and
sinceany impressedforcemust in effect react againstthe earth, it is
helpful to assumethat every masshas two terminals,one of which is
somearbitrary point of the mass,the other beinga point near the mass,
whichis fixed to the earth, as shownin Fig. 3. Here againthe velocity
differenceacrossthe mass(relative velocity) of its terminalsis consideredpositiveat the left terminal when the right terminalis moving
to the right. (Whilein ordinarylanguage
we speakof thevelocityofthe
massinsteadof the velocitydifferenceacrossthe mass,this is similarto
the caseof any two terminal electricalelement,one side of which is
grounded,wherein the potential of the high side is the same as the
potential differenceacrossthe element.) If we impresson a massan
externalpositiveforcef' as shownin Fig. 4, the masswill receivean
acceleration
whichwill be positiveat the left terminal andwhichwill
call forth a reactive tensile force between the terminals of the mass
• Anyunaccelerated
frameof reference
woulddoequallywell.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
F. A. FIRESTONE
'/
/
I
I
•
,•' [
/
/
/
253
/
/
FIG. 4.
A tensile force tending to pull these terminals
together.
A compressiveforce tending to push these terminals apart.
-
A springin tension.The displacementdifferences
must therefore be positive at the left terminal,
meaning that as seen from the left terminal the
right terminal must have moved in the positive
direction. Springlengthened.
r
A spring in compression.
The displacementdif-
ference
is therefore
negative
at theleft terminal.
Springshortened.
_•
r: must
Resistance
inhave
tension.
therefore
the The
signvelocity
shown, difference
resistance
-
v q-"
-
growinglonger.
r
-
Resistancein compression,
growingshorter.
/•7•
..... Mass
with
acceleration
difference
of
signs
shown
m must be in tension.The fight terminal is accelerating toward the right.
must be in tension.The left terminal is accelerating
to the left.
Fro. 5.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
254
Jovs•x•.OrTHEACOUSTICAL
SOCIETY
[January,
A andB, ofmagnitude
f--ma. Thusthetensile
force
ft hasproduced
a
tensileforcethroughthe massto the earth.4It is asif the earth4did not
liketo seemasses
accelerated
andactedon themwitha restraining
force.Assuming
as beforethat a tensileforcethroughan elementis
positive,
that is, theelement
is tryingto pullits terminals
together,
thereis a positiveforcethrougha masswhenthereis an acceleration
acrossit whichis positiveat the left terminal. Similar conventionscan
beworked
outfora moment
ofinertiain a torsional
system.
The aboveconventions
aresummarized
andexemplified
in Fig. 5.
Mechanical
elements
willbeassumed
to beconnected
mechanically
in series
whentheyarejoinedendto endasshown
in Fig.6. In suchan
arrangement
isit obvious
thattheforcethrough
alltheelements
isequal;
if oneelement
isin tension
theyareallin tension.
Likewise
thevelocity
difference
across
a series
ofelements
isthealgebraic
sumofthevelocity
Fro. 6.
differencesacrosseachelement. When a number of mechanicalelements
areconnected
to twocommon
junctionpoints,asshownin Fig. 7, they
maybesaidto beconnected
in parallel.Theconnecting
barontheright
isassumed
to remainvertical.It isobvious
thatin suchanarrangement
the velocity differenceacrossall of the elementsis the same.Further-
morethe total forcethroughthe combination
is the algebraicsumof
the separateforcesthroughthe elements.
Springsand resistancesmay be connectedeither in seriesor in
parallelbut a numberof masses
canbe connected
in parallelonly,as
oneterminalof eachmassis connected
to the earth.Two masses
only
maybe connected
in seriesthroughotherelements
asshownin Fig.6,
but if theywereconnected
directlytogether,theywouldin effectbe in
parallel.Of course,
springs
andresistances
may be connected
eitherin
seriesor in parallelwith masses.
We may statetwo additionalrelationswhichhold for any linear
mechanical
system,the forcelaw and the velocitylaw. According
to
theforcelaw,thealgebraic
sumof all theforces
actingonanyjunction
4 Or other frame of reference.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. F•tv.s•o•v.
255
of mechanical elementsis zero. For instance if we considerthe central
junctionin Fig.7 andassume
thata forcewhose
arrowisawayfromthe
junctionshouldbe countedpositive,then
f--f•--f•--f•--f4=O.
The velocitylaw statesthat the algebraicsumof the velocitydifferencesaroundany closedmechanicalcircuitis zero.It is necessary
to
take as the signof eachvelocitydifference
the first signwhichis seen
onapproaching
theelement
in goingaroundthecircuit.Thusin Fig.6
if the signsof the velocities
areassumed
asshown,
v• - v• + va - v4 = 0.
Alsoin Fig. 10thevelocitylawappliedaroundthecentralcircuitwould
give v•+v4+v•-v•=O.
Fro. 7.
We follow the usual conventionin definingmechanicalimpedance
asthecomplex
quotientof theforcethroughandthevelocitydifference
across an element or combination of elements.
(This definitionwasoriginallychosenwith the conventional
analogy
in mind.) In a seriesof mechanical
elements
havingindividualimpedancesof •, z•, zs,etc., the impedance
of the combination
wouldbe
f
• + ,• + • + +
1
•
•
1
•
/+7+7++
1
1
1
-+-+-++
Thus mechanicalimpedances
in seriesadd as the reciprocalof the
sumof the reciprocals
asdo electricalimpedances
in parallel.Similarly
if we have a numberof mechanicalimpedances
in parallel,the impedance of the combination is
So mechanicalimpedances
in parallel are additive like electricalimpedances
in series.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
256
JOURNAL
O1•THEACOUSTICAL
SOCIETY
[January,
In the above descriptionof the mechanicalelementsand the mechanicalcircuit no detailedassumptionhasbeenmade as to the nature
of any analogywith the electricaldrcuit which may be noted later, although someof the conventionswere suggestedby analogy. By consideringeach massas having one terminal fixed to the frame of reference, every mechanicalsystemis reducedto a multiplicity of dosed
mechanicalcircuits,thereby preparing•he way for the applicationof
electricalanalogies.
THE CONVENTIONAL ANALOGY
The conventional mechanical-electricalanalogy may be derived
from the fact that for most mechanicalsystems,an electrical system
can be invented of sucha sort that the differential equation• of motion
in the two systems,as expressedin termsof displacementand charge,
respectively,will be of the sameform. If impedancesare definedas in
the followingvectorial equations
= f/z (mechanical) I = E/Z (electrical)
the form of the impedances
as derivedfrom the differentialequations
will be similar and will justify the following conventionalanalogy.
THE CONVENTIONAL ANALOGY
Electrical
Mechanical
Forcethrough=f (dynes)
Velocityacross--v (cm/sec.)
Displacement
across--s (cm)
Impulsethrough--p (dynesec.)
Impedance
= • (ohms)*
Resistance
= r (ohms)
Reactance• x (ohms)
Mass--m (grams)
Compliance-½(cm/dyne)
Power=f • (ergs/sec.)
• of resistor--r (ohms)
• of mass=io.n (ohms)
z of spring=-i/o,½ (ohms)
Impedancesin series
e.m.f. across--E (volts)
Currentthrough--/(amperes)
Chargethrough=Q (coulombs)
Voltage
impulse=fEAt
(voltsec.)
Impedance=Z(ohms)
Resistance
= R (ohms)
Reactsnee
= X (ohms)
Inductance--L (henries)
Capacity=C (farads)
Power= E1 (watts)
Z of resistor--R (ohms)
Z of inductance=i•oL(ohms)
Z of condenser=-i/•oC (ohms)
Impedancein series
1
11zt+ 11z,+ 11z8+
Impedancesin parallel
Z = Z• + Z, + Z, +
Impedancesin parallel
1
Z-
1/& + i/z, + i/z, +
Kirchhoff's laws
Forceand velocity laws
Sum of velocity differencesaround closed Sum oI currentsto a •unction is zero
circuit is zero'
Sumof forcesto a junctionis zero
Sum of e.m.f.'s around a mesh is zero
* Mechanicalohm=dyne/kine.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. Fmv.s•o•v.
257
Thus in the familiar problemof the forcedvibration of an elastically
boundmasswith friction, and its analogouscircuit, shownin Fig. 8, the
differential equationsare
mdas
rds cs = feio'
t Ld•Q
RdQ
Q Eei'ø
dta + dt
C
at' +•-+
+--=
"
The steady state solutionsof thesein vector forms are
f
r + i
f
E
corn
R + i
E
col
Therefore the velocity acrossthe mass due to the impressedforce is
the same as the current through the inductance due to the e.m.f.E.
_/
Conventionalanalogue
Mechanicalsystem
FIG. 8.
However, while the mechanicalelementsare connectedin parallel,
the analogouselectricalelementsmustbe connectedin series.This must
alwaysbe the casewith this analogysincemechanicalelementsin
parallelhave a commonvelocitydifferenceacrossthem and electrical
elementsin serieshave a commoncurrent through them. In general,
mechanicalelementsin seriesare representedby electricalelementsin
parallel, and viceversa,so that in a complicatedmechanicalsystem
the analogouselectricalelementsmust be placed in a manner quite
contraryto intuition. A mass,which always has one terminal on the
earth, will usually be representedby an inductancein the high side of
the line, and a springwhichis in serieswith the mechanicalcircuit and
must transmit all the force from one part of the mechanicalsystemto
another
will be shown as a condenser across the line with
one side
grounded,while a springwhichconnectspart of the mechanicalsystem
to the earth will be representedby a condenserin serieswith the high
wire. This is shownin Fig. 12.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
258
JOU•,NAI.
O• •HE ACOUS•C^•.
SOCiEtY
[January,
The difficultyis alsoindicatedin the useof the words"thr.ough"
and "across"
in the tableof analogous
quantitiesabove.Forcehasa
"through"character
like current;the forcethrougheachof a series
of
mechanical
elements
is thesame,justasthe currentthrougheachof a
series
of electrical
elements
is thesame.Also,velocitydifference
hasan
"across"
characterlike potentialdifference;
the velocity
difference
acrosseachof a numberof mechanical
elementsin parallelis equal,
just as the potentialdifferences
acrosselectricalelements
in parallel
are equal. It is in ignoring these fundamental characteristicsof the
analogous
quantitiesandplacing"forcethrough"analogous
to "e.m.f.
across"
that the conventional
analogy
becomes
unnecessarily
difficult
of application.
Furthermore it is noted in the table that the laws for the addition
of impedances
are not analogous,
and alsothat thereis a lack of accurate correspondencebetween Kirchhoff's laws and their mechanical
analogues.
THE NEW ANALOGY
The analogyhereproposed
may be derivedfromthe similarityof
the equations
v=f•
E=
IZ
where• is the bar impedance,the reciprocalof the mechanical
im-
pedance.
The realpart of • is the bar resistance
•, andits imaginary
part, the bar reactance
•. Sincethe bar impedance
of a springof compliancec is icocand the impedance
of an inductance
L is icoL,the impedanceof an inductance
L =c is at all frequencies
equalto the bar
impedance
of the spring.Similarlythe far impedanceof a mass
m is -i/cornand the impedance
of a condenser
is -i/coc sothat the
impedance
of a capacityC =m will at all frequencies
be equalto the
bar impedance
of the mass.The bar impedance
of a mechanical
resistance
r ohmsis 1/r andis thereforeequalto the impedance
of an
electrical
resistance
R = 1/r ohms.In thiswaythe following
analogyis
established.
THE NEW ANALOGY
Mechanical
Forcethrough=f (dynes)
Velocityacross=v(cm/sec.)
Displacement
across--s (cm)
Impulsethrough-p (dynesec.)
ßBar impedance-- • (ohms)*
Electrical
Current through--I (amperes)
e.m.f. across-E (volts)
Voltageimpulseacross=fEdt(voltsec.)
Chargethrough--Q (coulombs)
Impedance= Z (ohms)
Bar impedance--velocity
across/force
through.
ohms
-- barohms--kines/dyne.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
F. A. FIRESTONE
259
Tax N•.w ANALOGY
(Coni'd)
Electrical
Mechanical
bBarresistance=
• (o-h--•ms)
øBarreactance=• (ohms)
Mass=m (grams)
Compliance
= ½(cm/dyne)
Power=fv (ergs/sec.)
• of resistor=• oh•)
• of mass=-i/o•m (ohms)
• of spring=io•c(ohms)
Bar impedancesin series
Resistance
= R (ohms)
Reactance= X (ohms)
Capacity= C (farads)
Inductance= L (henries)
Power= IE (watts)
Z of resistor=R (ohms)
Z of condenser=-i/o•C (ohms)
Z of inductance=io•œ(ohms)
Impedancesin series
Z = Zx + Z• + Z3 +
Bar impedancesin parallel
Impedancesin parallel
1
1
Z-
1/Z• .-I-l/Z, .-I-1/Z, .-I-
1/• + 1/•, + 1/•3 +
Force and velocity laws
Sum of forces to a junction is zero
Sum of velocity differences around closed
Kirchhoff's
laws
Sum of currents to a junction is zero
Sum of e.m.f.'s around a mesh is zero
circuit is zero
Bar resistance=real part of bar impedance.
Bar reactance= imaginarypart of bar impedance.
It has seemedadvisableto introduce a new term, bar impedance,
whichis equalto velocityacross/force
through.It is natural that the old
analogists,having arrived on the ground first, should have chosento
define impedanceas force/velocitysince that fitted in with the other
assumptionsthey had made. But in the author's opinion, all of their
assumptionswere unwise and led to the left-handed result that while
electricalimpedancesin seriesare additive, mechanicalimpedancesin
seriesmust be added as the reciprocalof the sum of the reciprocalsas
was shownabove. It would have been better if this new analogy had
beenthought of first, for in that casethe quantity which we have been
forcedto call "bar impedance"wouldhavebeencalled"impedance"and
wouldhave beensubjectto the samelaws of addition as are found in the
electricalcircuit. It is now too late to changesuddenlythe unfortunate
definitionof impedancewhich has beenso much usedin the past, soit
is recommendedthat the term "bar impedance"be used. Then if the
new analogyshouldprovepopular,the time may comewhen the old
definitionof impedancewill have fallen into disuse,at which time the
"bar impedance"may be shortenedto "impedance"with the new deftnition. •
• Of course,bar impedanceis the same as mechanicaladmittance as usually defined,
but to use the latter term would spoil the analogyand causeconfusion.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
260
JOURNAL
O•'TIlE ACOUSTICAL
SOCIETY
[January,
This nomenclaturetherefore prepares the way for an evolutionary
changeof definition.In the meant•ime• will indicatethat this quantity
is z below the line, that is, the reciprocalof z as ordinarily defined.It
shouldbe noted,however,that s is the real part of • andis equalto r/z •,
being the reciprocalof r only when z--r, a pure resistance;similarly
for •.
The unit in which bar impedanceis measuredis the bar ohm (ohm)
which is one kine per dyne. In time the bar might be droppedthereby
constitutinga changeof definition.
In a seriesmechanicalsystem,the bar impedanceof the combination
is the sumof the bar impedancesof the elements.
v• + v•. + v• +
• =
...
=•+•.+•a+'''
ß
f
Similarly for a parallel mechanicalsystem
v
f• + f•. + fa + '''
1
1/• + 1/•. + 1/•a + ...
Thus the bar impedancefollowsthe samelawsof addition as electrical
impedance.
The new electricalanaloguefor displacementis the voltageimpulse,ø
(fEdt) thisbeingthe electricalquantitywhichis measured
by a ballistic
galvanometer.It was given this name by the old analogistsbecauseto
them it was analogousto mechanicalimpulse.The lack of correspondenceat this point in the new analogyis again a mere matter of definition. In any problemwherea simpleharmonicdisplacementamplitude
is specified,it is convenientin the new analogyto convertit to velocity
by multiplying by ico.
In both analogies,mechanicalpowerand electricalpower are analogous.
It will be notedthat in this analogythe words"through"and "across"
are analogouslyused.In the old analogyit was appealingto have force
and electromotive force be analoguesbecauseof the similar sound of
the words which, when looselyused, appear to have a similarity of
meaning; but when one analyzes the casemore closelyand finds that
force throughis analogousto electromotive force across,some of the
attractivenessis lost. One may also favor the old analogyif he thinks
of e.m.f. as causeand current as effect, of forceas causeand velocity as
• Bennet and Crothers,IntroductoryElectrodynamics
for Engineers,p. 470.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. Fm•.s•o•.
261
effect.However,in • serieselectricMcircuitit is moreconvenientto
considerthe currentthrough•n element•s the c•use•nd the e.m.f.
•crossthe element•s the effect,•nd in • seriesmech•nicMsystem,of
forcethrough•n element•s c•use•nd velocity•crossthe element•s
effect,therebyin that c•sef•voringthenew•nMogy.
With this •nMogyMso,the differentiMequations
of motionin the
•nMogous
systems
will be of the s•meformif writtenin •ppropri•te
v•ri•bles.For instance,consider
the problemin Fig. 9 where• velocity
!
/
!
'/
!
I
!
/
I
!
FIG. 9.
ve• is impressed
in serieswith a resistance,
spring,andmassandwe
wish to find the resultingforcethroughthe system.Applyingthe
velocitylaw aroundthe mechanical
circuitand usingthe impulse
p--ffdt as dependent
variablewe get the followingdifferential
equations'
c•
d•'p
4-----p
•'ø•
-• • dp
= ve• ; Ld•Q
+ RdQ
+ •Q= Ee
ß
dt•'
'•
m
dt•'
dt
C
Sincef=dp/dt andE =dQ/dt the steadystatesolutions
of theseequations in vectorial
form are
v
v
-- 1loom)
f= • 4-i(coc
- • ; I =R +
E
E
- ß
i(ooL-- 1/ooc) Z
The velocityacross
any elementcanbe foundby multiplyingthe
abovevalueoff by the bar impedance
of the element.
In the abovedifferential
equations
andtheirsolutions
theanalogyas
set forth in the above table is obvious.In the seriesmechanicalcircuit
shown,the impulse
througheachelement
is the samewhilethe displacements
across
themaredifferent,
consequently
it ismoreconvenient
to writethe differential
equation
with impulseasdependent
variable,
thandisplacement.
Wemaytherefore
saythatif thefounders
ofvibrationtheoryhadhappened
to consider
theforcedvibrationproblem
of
the seriesmechanical
system
with impressed
velocity,insteadof the
parallel
system
withimpressed
force,theanalogy
herepresented
isthe
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
262
Joussaz or Tar. ACOUSTICAL
SOCIETY
[January,
one which they would have noted as being the most obviousand appealing.
EXAMPLES
The abovedifferentialequationshave beenpresentedfor the benefit
of thosewho feel that an analogyshouldcome from a similarity of
differentialequations.However, when one wishesto find the steady
state e.m.f.'s and currentsin a given electricalcircuit he doesnot write
down any differential equations.He finds the impedancesof the elements at the given frequencyby well-knownformulae,appliesKirchhoff's first law to the currentsenteringthe junction points, applies
Kirchhoff'ssecondlaw to the voltagesaroundeachmesh,and solvesthe
simultaneousequations thus derived for the unknown currents and
voltages. It is, therefore, desirablethat one should not have to write
the differentialequationsfor either the mechanicalor the electricalsystem in findingthe electricalanalogueof a givenmechanicalsystem.
In the new analogy, the equivalent electrical circuit is drawn to resemblethe originalmechanicalsystem,rememberingthat one terminal
of eachmassis connectedto earth. Each springis representedby an
inductanceL=c; each massis representedby a capacity C=m one
sideof which is connectedto ground;eachmechanicalresistanceof bar
resistance• is replacedby an electricalresistanceR =•. Mechanical
elements in series are representedby electrical elements in series;
parallel mechanicalelementsby parallel electricalelements.If there is
any questionas to the validity of the equivalent circuit, Kirchhoff's
laws may be applied to the junctions and meshesof the circuit and
theseshouldyield equationswhichare equivalentto thosewhichwill be
obtained on applying the force law and the velocity law to the mechanicalsystem.Kirchhoff'sfirst law for the currentsflowingto any
junctionis quite analogousto the forcelaw for the forcesactingon any
junction; Kirchhoff's secondlaw for the e.m.f.'s around a mesh is
analogousto the velocity law for the velocitiesaround a closedmechanical circuit.
Having arrived at this much of an understandingof the mechanical
problemwe might considerabandoningthe analogy and working the
mechanicalproblemdirectly by an applicationof the forceand velocity
laws.In caseit is necessaryto write downKirchhoff'sequationsfor the
electrical circuit and to solve them, little is gained by drawing the
equivalent circuit; the analogousequationscan be written down at
oncefor the mechanicalsystem.Many well-knownelectricallaws as,
for instance,the reciprocaltheoremand Thevenin'stheorem,can be
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. FIRESTONE
263
convertedby the analogyinto equivalentmechanicallaws and used
directly. In this manner, the electrical techniquefor the solutionof
problemscan be usedwithout convertingthe problemto an electrical
circuit. In casethe propertiesof the analogouselectricalcircuit have
alreadybeenworkedout sothat no equationsneedbe written or solved,
then there is a decidedadvantagein working with the electricalanalogue.
v
o00-c'
?,
Fro. 10. M•chanical system.
FIo. 11. N•
an•ogu,.
La•ma
FIo. 12. Conv,ntionalandogu,.
As an exampleconsiderthe linear mechanicalsystemshownin Fig.
10 in which an impressedvelocity is operatingand we might wish to
know the velocity across(or of) mo.The new analogue,derivedas explainedabove,is shownin Fig. 11, and the electricalcircuitlooksmuch
like the mechanicalsystem. The e.m.f. acrosscowill be the desired
velocity acrossmo.The conventionalanalogueis shownin Fig. 12 and
the circuit is just the oppositeof what onewould at first glanceexpect.
C• and R• mustbe connectedin parallelbecausec• and • havethe same
forcethroughthem. Lamustbe in serieswith the highwire eventhough
mahas one terminal to earth becausethe forcethrough mais the differ-
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
264
Jo•J•• o• •a•. Acovs•c^• Soc•.•
[January,
encebetweenthe forcesthroughc4and c•. The currentthroughLo is
the desiredvelocity acrossmo.
Or suppose
wewishto solvethewell-known
problemof the forced
vibrationof a mass,spring,andresistance
in parallel,asshownin Fig.
13,wishingto findthevelocityacross
themass.Thenewanalogue
is a
1•
F
L
c
New analogue
Mechanical system
FIG. 13.
constantcurrentgeneratorparalleledby a resistance,
inductanceand
capacity,and our analogous
problemis to find the e.m.f. acrossthe
condenser C.
1
E =IZ
=I
I
=
+
ß
+ i(ooC-
Now putting in the mechanicalvaluesof the analogous
electricalquantities we have
1/• q- i(o.,m- 1/•c)
which is recognizedas being the correct result when we remember
that i is the reciprocalof the mechanicalresistanceas usually defined.
THE NEW ANXLOGUE OF ELECT•O•ECHANICXL
DEWCES
In treating electromechanical
devices,suchas microphonesand loudspeakers,by convertingtheir mechanicalparts to analogouselectrical
structureswhichare to be suitablycoupledto the electricalsystem,it
must be rememberedthat in the analogiesset forth above, the c.g.s.
unitswereusedon the mechanicalsidewhilepracticalunitswereused
on the electricalside. This resultsin power in ergsper secondbeing
analogousto power in watts, thus differingby a factor of 10•. As long
as we deal in analogiesonly, thereis no harm in having the powersin
the analogous
electricalcircuitbe 10• timesthe powersin the mechanical system;but when an electricalcircuit is coupledto an analogue
circuit, the energyleavingthe electricalcircuit must equalthe energy
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. Fm•.s•o•.
265
entering the analogue circuit. One way to resolve this difficulty is to
useasour unit of force(10')•/• dynes,analogous
to oneampere;and as
our unit of velocity (10')•/• kines, analogousto one volt. The use of
theselarge units will not changethe magnitudesof our mechanicalimpedances,which involve the ratio of force and velocity, but will cause
the power,whichis the productof forceand velocity,to be expressed
in
watts.
Considerfor example a generator connectedto a telephonereceiver
whose diaphragm may be idealized as a mass connected through a
spring to the receivercasewhich, for simplicity, is consideredas stationary. There is a two-way interaction between the electrical circuit
and this mechanicalsystem:a current through the magnet producesa
forceon the diaphragm,whoseamount in dynesper ampereis assumed
known; also any motion of the diaphragm generatesan e.rn.f. in the
magnet circuit, whoseamount in volts per kine need not be known.
This couplingbetween the electrical circuit and the analoguecircuit
can be representedby an ideal transformerwhoseprimary is in series
with the electricalcircuit and whosesecondaryis in parallel with the
condenserand inductancewhich representthe effectivemassand complianceof the diaphragm.The ratio of primary to secondaryturns is the
forcefactor of the magnetsystemas expressed
in large dynes((107)•/•'
dynes)per ampere.The resultingcircuit may be solvedin the usual
manner to find the voltage acrossthe condenser;this will be the velocity of the diaphragmexpressed
in largekines ((10') •/• kines).
If the electromechanical
couplingis electrostaticinstead of electromagnetic,then voltage in the electricalcircuit resultsin a force on the
mechanicalsystem,that is, it resultsin a currentin the analoguecircuit.
Also the velocityof the mechanicalsystemgeneratesa currentin the
electrical circuit. This calls for an inverse transformer to couple the
electricalcircuit and the analoguecircuit of sucha nature that the current throughthe secondaryis a constanttimes the voltage acrossthe
primarywhilethe currentthroughthe primaryis a constanttimesthe
voltageacrossthe secondary.This is not a familiar devicebut it can be
replaced by an ideal transformer if either the circuit connectedto its
primaryor to its secondary
be replacedby an inversenetworkwith respectto oneohm.* Changinga newanalogueto its inversenetworkwith
respectto one ohm is equivalentto a changeto the old analogue.In
this problemof electrostatic
coupling,the old analogyis advantageous
?SeeShea,Transmission
Networks
and WaveFilters,p. 311, or Johnson,
Transmission
Circuits
for Telephonic
Communication,
p. 230.
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
266
JOURNAL
Or TIlE ACOUSTICAL
SOCIETY
[January,
to the extent that it permits an ideal transformerto be usedin coupling
the electricalcircuit to the analoguecircuit.
TI•E ACOVS•ICA•.-E•.EC•ICA•.
ANALOGY
Acoustical transmissionsystemsare not analogousto mechanical
systemsin quite such a simple manner as one might at first thought
suppose.In such an acousticaltransmissionsystemas an acoustical
filter, eachmain conductingtube has as its electricalanalogue,an electrical line (two wire) havingtwo input and two output terminals.s If a
tube is used as a side branch with either a dosed or an open end, it is
as if the input terminalsof the equivalent line were connectedto the
main conductingline while the output terminalswere open circuited
or short circuited.
The two variablesusually'usedin the discussion
of an acoustical
transmissionsystemare the soundpressurep at a surfaceand the
volume velocity V through the surface.The acousticimpedance,Za,
on a givensurface,is definedas the complexquotientof p and V. Thus
from the similarityof the followingequationsthe conventionalacoustical-electricalanalogymay be derived.
v=•/z•
t=•/z.
However, by definingthe acousticalbar impedanceas V divided by p
we could also write
V=p•a
E=IZ
therebyobtaininga new analogy.
Between the two mechanical-electricalanalogiespreviously mentioned, it was possibleto make a choiceas to the one which was most
complete.This was possiblebecauseit is easy to tell when mechanical
elementsare in seriesand when in parallel; also a strict analogueto
the Kirchhoff relationswas found with the one analogyonly. But if a
number of acousticaltransmissiontubes terminate or originate in a
commonjunction point, there is no a priori criterion as to whether
they are connectedin seriesor in parallel within the junction. The case
is similar to a number of electric lines entering a commonjunction
box; they may be connectedeither in seriesor in parallelwithin the box.
If onefirst assumes
the conventionalanalogy,then he will say that the
tubesare in parallel within the junction becausethey are subjectto a
commonsoundpressure;but if he assumesthe new analogy, he will
s W. P. Mason, B.S.T.J. 6, 258 (1927).
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15
1933]
F.A. FIRESTONE
267
say that they are in series,for the samereason.There is somejustification for a preferenceof the conventionalacoustical-electricalanalogy
in that the sum of the volume displacementsto any junction is zero
analogousto Kirchhoff'ssecondlaw for the e.m.f.'s arounda mesh.On
the other hand, the new analogyseemsmore rational in that with it, a
tube having an openend is representedby an opencircuitedline, while
a tube with a closedend is representedby a short circuitedline; in the
old analogy theserelationsare reversed.
Thus for acousticalsystems,the two analogiesseemabout equally
good.
CONCLUSION
The conventionalmechanical-electrical
analogyis incompletein the
followingrespects'
I. There is'a lack of analogyin the useof the words"through"and
"across" which indicates a fundamental
difference in the nature of the
analogousquantities, for instance,force through and e.m.f. across.
II. Mechanical elementsin seriesmust be representedby electrical
elementsin parallelr•nd viceversa.
III. Mechanical impedancesin seriesmust be combinedas the
reciprocal of the sum of the reciprocalswhile electrical impedances
in series are additive.
IV. There is an incompletenessin the mechanical analoguesof
Kirchhoff's
laws.
The new analogyis free from the above difficulties,which makesit
much easierto apply and understand.By consideringeach mass as
having one terminal fixed to the frame of reference,every mechanical
systemis reducedto a multiplicity of closedmechanicalcircuitsthereby
preparingthe way for the applicationof the electricalanalogy.In the
new analogy, the equivalentelectricalcircuit can be drawn with ease.
Even thosewho have never usedany analogy and never intend to do
so,will be adverselyaffectedby the fact that the old analogyhasbeen
usedin making the definitionof mechanicalimpedance.This unfortunate choiceof definitionwill retard the developmentof vibration theory
and make it all the more necessaryto reply on electricalcircuit theory
by analogy.
The author nominates for oblivion the conventional
left-handed
me-
chanical-electricalanalogy.
I wish to thank Mr. L. D. Montgomeryand Mr. Earl Burnsfor their
valuable suggestions.
September12, 1932
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 129.2.40.52 On:
Mon, 03 Nov 2014 19:56:15