SCIENTIFIC ABSTRACT PUKHOV, G. YE. - PUKHOV, G. YE.

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• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001343610004-9
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: November 2, 2016
• Document Release Date: March 14, 2001
• Sequence Number: 4
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

PUKHOV, G. Ye., prof.doktor tekhn.nauk
Analog computor for bent rods suggeated by Corbett and Calvert.
Trudy RISI no.11:124-129 '5R. (MIRA 13:5)
1. Tapanrogskiy radlotekhnicheskiy institut.
(Elastic rods and wires--Electromechanical analogies)

PUKTIOV, G.Ye., prof., doktor tel-hn. nauk; ILI YENKO, O.V., kand.tekhn.naiik
Electric analyzers for beams and frames on solid eleastic basis
and rigid supports. Trudy RISI no-11:130-135 158, (MIRA 13:5)
1. Taganrogski5r radiotekhnicheskiy institut.
(Structural frames--Electromechanical analogies)
(Girders--E'Lectromechanical analogies)

SOV/144-59-1-2/21
AUTHOR: Pukhov, G.Ye.2 Dr.Tech.Sci., Professor
TITLE: Electrical Analogue of the Scalar Product of Multi-
Dimensional Vectors
PERIODICALs Izvestiya vysshikh uchabnykh zavedeniy7
Elektromekhanika, 19597 Nr 1, pp 11-12 (USSR)
ABSTRACT: In computing techniques it is often necessary to evaluate
the expressions of the type:
alxl + a2x2 + . - . . + anxn = b
where al and xi can be either negative or positive.
It is seen from Eq (1) that the quantity b represents
the scalar product of two n-dimensional vectors, that is
b = a.1 The product can be evaluated by means of a
simple electrical circuit shown in Fig 1. This consists
of two n-pointed stars whose star points are joined
ttrough the conductance go. The circuit obeys the
relationship expressed by Eq (3) provided all the switches
Card IT, 7 T"21 - - - TI n are in the right-hand position. If one
1/2 of the switches, for example IT2i is in the left-hand
position, the system is described by Eq e

SOA!/144-5 9-2-1/19
Pukh.:-,v- a@Ye- Do.-.tor of Te:hnical 5"---ier%ces. Professor
I YLE, 'fh@ [e-mail] of Electrical Nc-twop-Ics
for Integrating
Difference and Different.Aal Eauations
PERIODICAL; Izirv-stiya Yys6bilib ii,..hbbnykj*j zav.@Aeniy, Elcktromekhanika,
1959, Nr 2, pp 3 ... @LO (USSR)
ABSTRACT, A peculiart-ty of so-c-alled net-orks of the first kind is
that for given boundary condit*iens the distribution of
voltages (currents) in @.he network is determined only by
the detailed strucA-ure of the modelling system and no
operatc-r jntL-r-Ve---1Tiori ii required@ This is because the
original set of equations resembles very closely those of
thk@ network itself. Where this is not so, as in Eq (1)
for exampla, tha methed encounters difficulties. Por such
an expression (an algebra!,--), riEt*works of the second kind
musT be used. A Itypictal ins+anc,-t ij a root-solver (Ref 5).
The present paper examines some theoretical topics and
illustrates the solution of ccmparatively high order
equations for v-ar�ous boundary conditions. Figure 1 shows
a C@ertain function y (.x* -) and the simulatin-- network
Coll;--. i stin-, of a number of voltage dividErs. Ths solution

68i 225
sov/144.-59-2-1/19
T11e Construction of Ble;L7trir:al Networks for Integrating Difference
and Differential Equations
of the equation f[y(x)l= 0 may be found using a syst
like the bloak-diagram of Figure 2.' The block :f[Y(XT
is a scalar proaket analogue (AC-n in the original or, ere,
SPA) @onsis-Ling, in the linear case, of ohmic elements and
de5cribed previously by the author (Ref 4). The block OC
iF a feedbank devIce. The entire loop forms a self-
adjusting system. Suppcse ihe linear difference equation
(5) is given and a solution isxGquired for the initial
conditions of Eq (16). The solution depends on Euler.'s
algeritbm cr, what is known in engineering as the method
cf suncessi-ve ordinates. For the sake of definiteness the
--arresponding practical circuit of Figure 3 assumes n = 2
The basic comq@,nents are: 1) the network y on which
the @t equirea fonction y(h) j.,; found as a reqult of the
a, i (-) I t.@@ the network setting, tip the function
t: -'PA raprii-enting Eq (5); 4) the null-
NI -ay' registering the deviat-ion e(k) in
(0-11 for rL = 2 3 5) -@-ommutator K with four moving
T-he prc---@.F-dure is as follcwa: a) sat u

68- 09
SOV/144-59-2-1/19
11'ae %Con!@itruction of Natworks for Integrating Difference
aiid. Di f f er eiit i al Equat ions
I.Z) on b) set up a 13 (o) (0). a 2(o) and N
on the SPA! c) sst initial yol Y1 on y
d) divid-er 2 on y is adjusted to give zero-.indication
(this gi.va_z yl?, e the moving r-onta--.ts on K are
di.splac.-ed one 3tap to the right and a 0(1), aft), a 200
are s:-@t on the SPA: f) divider 3 on y is set to restore
zero--indication, th-as finding y- . Operations e) and
f) are car-ried out again on dividers 4. 5, 6. .... finding
, et---. These operations could be automated.
Y4 9 Y5'J '@r6 .
The use of a digital voltmeter would enable the bulk
of y -co be reduced. The arrarigement also simplifies
when the coeffi;..ients of Eq (5) are r;onstants (independent
of' k ) . The question of ac-curacy requires special
consideration and it is merely stated that the @rrox- is
redu@_ed the liigher the sensitivity of the null indicator,
Catd5/5 t-he niore a-.-,,@urately the inltial values and a W are
set in. The use of the in-regrator is then deecribed for
VrI,

6 8129
SOV/1 44-59-2-1/19
Th(.@ C0,1St-t-LtC-t-.L()n Of E.It-Aric-al NeLworks for Int6grating Difference
and Differential Equations
solving the difference equatior (5) within the interval
0 :Ck _0 Q n)
where Ej = 1 while all the remaining electromotive
forces (as well as the current sources) are equal to 0.
If the quasi-negative resistances are realised by means
of the current sources (see Fig 2), the system can be
described by (Eq 9). For this case the condition
sufficient for the convergence of the balancing process
is expressed by Eq (10). If the above is checked
experimentally it is more convenient to employ
i-n
21iji lid > 0, Q = 17 ..... I ,)I

69198
3/144/60/000/02/002/019
E192/9182
The Conditions of Convergence for the Balancing of an Electrical
Network containing Quasi-Negative Resistances
where Ij = 1, while the currents of all the remaining
current sources are equal to 0. In a practical
application of the networks with quasi-negative
resistancesq it may be necessary to replace one type of
negative resistance (with a voltage source) by another
type (with a current source). In order to determine if
such a transformation effects the convergence of the
balancing process it is necessary to determine the
relationship between the coefficients Pjj and qij.
The circuits for determining these coefficients are
illustrated in Figs 3 and 1+. The relationships between
the coefficients take form of Eqs (12) (13), (14) and
(15), where kii is defined by Eq (163. From the above
equations it is"concluded that the replacement of the
resistances with voltage sources, by the resistances with
current sources, or vice versa, does not influence the
convergence of the balancing process. The above methods
of balancing can be secured by minimising the sum of
5/6 moduli of the deviations or the sum of the squares of V@

69198
Z+4/60/000/02/002/019
B192/9182
The Conditions of Convergence for the Balancing of an Electrical
Network containing Quasi-Negative Resistances
deviations. The sum of the moduli of the deviations can
easily be produced by employing a set of rectifiers and
an adding circuit; the resulting sum is then expressed
by Eq (18). The above balancing process Is illustrated
by a practical example; the actual network (which
analogues a beam) is shown in Fig 5. The circuit is
described by Eqs (19) and (20) from which it follows that
the coefficients Pij are given by Eq (21).
I-ard There are 5 figures and 10 references, of which 2 are
English and 8 Soviet. Two of the Soviet references are
translated from English.
-.--70N: Kafedra teoreticheskikh osnbv-el4ktftSOkhniki7
Kiyevskiy institut grazhdanskogo voZaiihnogo flota
(Chair of Theoretical Fundamentals of Electrical
Engineering,.Kiyev Institute of Civil Aviation)
December 7, 1959

69408
s/144/6o/ooo/o4/003/017
14, 4no E194/E455
AUTHOR: Pukhov, G.Ye., Doctor of Technical Sciences, Professor
TITLE: The Analogue Representation of Three-and Five-Term
Equations of Structural Mechanics
PERIODICAL: Izvestiya vysshikh uchebnykh zavedenly, Elektromekhanika,
1960, Nr 4, PP 17-19 (USSR)
ABSTRACT:
Card 1/3
Many problems of structural mechanics involve the solution
of three- and five-term algebraic equations. The three-
term equations are of the form of Eq (1) with
coefficients satisfying the conditions given by Eq (2);
the five-term equations have the form of Eq (3), whose
coefficients satisfy the conditions of Eq (4). Articles
by other authors have shown that the system of equations
(1) with conditions (2) may be represented by an
electrical circuit consisting of 3n - 2 ohmic resistances.
When some or all of the auxiliary coefficients are -
positive, the circuit has twice the number of loops than
when all are negative. The present article shows that
Eq (1) may be represented by an analogue which contains
not more than 2n - 1 resistances and examines the
conditions under which the analogue may also be used to

69408
s/144/6o/ooo/o4/oO3/0l7
E194/E455
The Analogue Representation of Three- and Five-Term Equations of
Structural Mechanics
find the roots of five-term equations by a method of
successive approximation. If all the auxiliary
coefficients of Eq (1) are negative, the analogue
circuit is of the form shown in Fig 1. It is
demonstrated that, by appropriate reversal of current
at particular points in the system, this same circuit
may be used to represent Eq (1) when the auxiliary
coefficients have other signs. The circuit for the
case when all the auxiliary coefficients are positive is
shown in Fig 2 and Eq (8) apply. Eq (1), (6) and (7)
may then be used to determine the voltages, currents and
conductivities of the circuit. If some of the
auxiliary coefficients are negative and others positive,
combined circuita% may be formulated: an example of th,is
kind is shown in Fig 3. The conditions under which
electrical analogues of three-term equations may be used
to solve five-term equations are then considered.
Eq (3) is rewritten in the form of Eq (9), that is in a
Card 2/3 three-term form. The method of successive approximation
q___

69408
s/144/60/000/04/003/017
E194/E455
The Analogue Representation of Three- and Five-Term Equjations of
Structural Mechanics
is then explained. It is shown that the five-term
equations can be solved provided that the moduli of the
main coefficients are greater than the sum of the moduli
of the auxiliary coefficients in each line of Eq (3).
This condition is frequently fulfilled in calculations
on various thin-walled or rod structures. There are
3 figures and 5 Soviet references.
ASSOCIATION:Kiyevskiy institut grazhdanskogo vozdushnogo flota
(Kiyev Institute of Civil Aviation)
SUBMITTED: September 20, 1959
Card 3/3

4 g-0 0
AUTHOR. Puldlov) G,,Ye, (Doctor
_F
TITLE: Some Design Finciples
wi-th Displ'ac@ed Joi.nts
82918
s/144/60/000/006/003/004
EO4l/E12l
of Technical Sciences, Professor)
for Electrical Models of Frames
PERIODICALs. Izvestiya vysshikh uchebnvkh zavedeniy,
Elekt:romekhanika, 19601'5/No 65 pp 1-7-2:7
TEXT,- Previously the use of the EMSS-1, -5 and -6 mach-ines
for .51mulatLng s-t*.,ucTura_1 problems of this kind has encountered
fi,:,ul ties because orilly part of the task c@ould be loaded; the
rest was a manual calculation. The present. paper describe's hl@
methods avoiding this disadvantage, Fig 1 shows the general case
o."' a freely supparted. rod subject to transverse 1..--)ads at ths-1
support@, bendi -ng moments at either end and a [e-mail] '
az,bit*@ar.I.ly distributed external loading. The -relations between
th,-. mean and. t.:,rminal slopes and the force-s and. moments are Eqs M,
(2) and U). Wben the ends of the rod are rigidly fixed the
equat.1o.ris are Eqs (Lt-), (5) and (t). The lfreel and 1--estrained'
equations are c,:ombined in Eqs M@ (8) and (9). It is pointed out
at th.1.3 stage that; si-mallatJon is greatly faC.1.11tated by wo-rking in
t9rm.:_:- of either sum or d1ffererines of bend..ing momenta a-z 1.n
Ga@.-d 1/3

82918
b/144/60/000/006/001/004
E041/E121
Some Design Principles for Electrical Models of Frames with
Displaced Joints
Eqs (9), (10). The equations may be generalized. as in Eqs (16),
(17), to allow for the moment of inertia of the cToss-section to be
,variable along the length of the rod. It is also shown that these
latteT equations can cater for longitudinal loadings as well as
continuous elastic support,, The acti-4e T-circuit of Fig 21 --is
desc-.ribed in Eq (2;) and can be used to simulate the structural
problem if' the resistance r is allowed to be negative. This
disadvantage may be overcome by three distinct methods*, adding a
bridging element to the T--circuiti Incorporating conditional.
voltage or cur:-ent sourc:es which enable a positive resistance to
appeai: negativE,-, cha--riging the form of analogue. FigE 3 and
together with Eqs (30) and (33) show how the second. method is used,
The first method has been described in Ref 3. The preferred
approac,,h is to 11,@ase the analogy on the rl-circtuit of Fig 5@ The
relevant set of identities are Eqs (41), (42')@ If a structure is
made up of a number of members then the circuit analogue iS
similarly built up wi.th circuit nodes co.r-responding to st-ruotural
Ca -i d 211

Some Design Prine.Iples for
Displac.ed Joints
82918
s/i L4/6o/ooo/oo6/ooVoo4
E041/El-i
Electrical Models of Frames witri
joints,, The twc-storey, single-span frame of Fig 6 is represented,
by the circuit of Fig 7 when T- circuit, analogues are used, and by
Fig 1.1 when the r, -form is used. In order to work In terms of
moment-differences the expression of' Eq (LF7) must be represented
using the sub-circuit of Fig 8, A more elegant solution is t1he
11warp-carculit" of Fig 10. The methods described have been used on
the EMSS-7 mac-,hine at; the computing centre of the Ukr-,SSR AS,
I
There are 11 fAgures and 10 refe:rencess: 8 Soviet and 42' Englisli,.
ASSOCIATIONt Kafedra teoretichesk.ikh osnov elektrotekh-nik.2.
Kiyevskiy institut grazhdanskogo vozdushnogo fiota
(Department of Theoretical Electrical Engineering,
Kiyev Institute of Civil Aviation).
SUBMITTED: No,.@rember 11, 195
,9
CaPd Y.7.@

FUKHOV, Goorgiy Yevgenlyevich, doktor takhnichookiy nauk-, prof.
Simulation of girders by means of balancing electric netse
Izv. vys. ucheb. zav.; elektromekh. 3 no.9:6-10 160.
(MIRA 15: 5)
1. Zavedu-yushchiy kafedroy eticheskikh osnov
elektrotekhniki Kiyevskogo i1ttuta grazhdanskogo,
vosdushnogo flota.
(Electric networks) (Girders-Nodels)

It
87988
'@'71900
S/144/60/000/011/002/008
E031/E255
AU"NJORS. Doctor of Technical Sciences- Professor
and Borkovskiy, B A , Poct--raduate Student
0
TITLE. On the Electrical Si.-aulation of a System of Linear
Algebraic Equations with Arbitrary Coefficient Matrix
0
PERIODICAL- Izvestiya Vysshikh Uchebnykh Zavedeniy Elektromekhan-
ika 1960, 4No 11. pp 36-37(
TEXT, It is usua-"iy assumed Ref. 1) that electrical
analo6-ues consistin@S of the elements R- L- and C can only be
constructed for systems of linear algebraic equations wi-th
symmetric matrices of coefficients while systems of equations
with unsymmetric matrices can be simulated by electrical circuits
only if transformers and amplifiers are used Refs 1, 2. 3).,
The present article aims to extend the domain of usefulness of
models for symmetric algebraic systems constructed from the
-elements R L and C.- -and to show how to apply them to the solution.
of systems of algebraic equations with arbitrary maturices. In
order to solvp the system AX - F with arbitrary matrix A on a@-
e2t--ctraca1 model- an electrical analogy of the following algebrai,-
Card 1/5

87988
S/144/60/000/0-11/002/008
E031/E255
On the ElectrLca--( Simulation of a System of Linear Algebraic
Equations with Arbitrary Coefficient Matrix
system Q A 7
--J I
must be constructed X and F are the vectors of the unknowns and
right hand sides of the system A" is the transpose of A, P
is an arbitrary symmetric matrix and Z is a vector of spurious
(undetermined) unknowns-, This is always possible since the matrix
of the system (2) is symmetric and no other restrictions are
imposed on systems simulated by electrical circuits containing R..
L and C elements (Ref 1), One of the possible methods of :on@-
structing an eLectr--cal circuit simulating the system (2) Is clear
Irom Fig 1 Fig I represents an electrical circuit described b-7
the followin@, system of 2n equations,
(Y +Y +Y -Y U +Y U +Y U +Y LT -+Y un@ ii
10@-Yli 12, lk 4 in) 1 11 1 12 2 lk k In
Car-I 2 , 5.

87988
S/144-/60/000/011/002/008
E031/E255
On the Electrical Simulation of a System of Linear Algebraic
Equations with Arbitrary Coefficient Platrix
(Y 4Y +Y .+Y Y U 4Y U Y U I
r,O nl. n2* nk- nn n nI 1 n2 2-t""'4Ynk k nk n@ r-
Y U +Y U -,-y U + +Y U +__4Y U'+(Yli+y + +Y )UCO.
11 1 1-2 2 13 3 Ik k 1n n 12 In
Y U 4-YU +Y U .,-Y U-4-(Y +Y (3)
n2 2 n3 3 nk k -n-n n- n1 n2
If @hr- equations Y +Y I'Y @, 2 1-Y ik"' "' +Y in =0 (i n)(u
are satisfijd then the system (3) can be written shortly as
F-
T-
10 Y
I Tj
- I .
MY-,"
By comparin,- eauations (2) and (5) it fol-Invis that rhe cir--uil
(FIF, 1) s-Imulates equations (2.), The elements a, of the matrix
A L@re -@imulated by the conductivities Y,k the co4onents Fi of
Ca-d

87988
S/144/60/000/011./002/008
E031./E255
On the Ele@: t r ical Sirioil@it ion of a System of Linear Algebr,,I 1 c
Equations with Arbitrary CoeffLcienl: Matrx
the vector of the ri-ht harid sides F are s:mu.lated by r-he current..S
S 0
i and the components x.,of the ve-tor of unknowns X are rep-
n@s,@_n@ed by the volta-es U The n-onducrivirlies of capacitors
C) k
simulate the positive elemZsn+ls of L,, and the ronductivIties of
,nductors SIMUlate the negative elements of A., The conductivitieS
Y are chosen so 'hat the "quations (4) are satisfied- i.e., so
the par-11-cular condiartivitimes of the nodes 17- 2'.
id@
N at whicl-i the current I re-oresenting the right hand st
of the eauations are introduc@d are zero, By measuring the
[e-mail] of the vertor a in amplittude and phase the values of
'he unkrovin vector X in E4uations (1) may be obtained in somr-
scale The r1rcuit (Fts,. 1) requires n r L and C elements to
sim,.ilat@ a system of the n-th order an arbitrary marr;x.,
There are 3 Sloviet references,
Kafedra reoreticheskikh. osno%, eIek--ro'kekhnik.,
Kievski-y institut grazhdansko,@_o flo-a
(De-jartment of Basic 'I'hecry Ele@:tric-n1l Enor-neer-
in- Kiev Institute of the Civ@l Flettz-
@Iard 4/,-;

87988
On "U'lle Elect-c-L.-'al o I a Sy3tue-@ o L-ine'al,
u j- J.
wi ry !@oefficJ entu
Eq u
.,u:-i t ico n s n
yfl
Yf f Yfz Y, n
- - -f--7
Y,, Y, 2IJ _Y?31
'C@
I n
v,
Y2
Y
Yll I '@n z
U,
ylo
y
'60
o
Card 5/5

PUEHOV.9--Georgi)~Yevgenlyevich, doktor teklin.nauk, prof.; SAMUSI, Vladimir
MMaylovich, aspirant
Electric circuits for integrating the equation yIV -2r2;j-v4y = q.
Izv. vys. ucheb. zav.; elektromekh. 3 no.12:20-25 160.
(MIRA 14:5)
1. Zaveduyushchiy kafedroy teoreticheskikh oanov elektrotekhniki
Kiyevskogo instituta grazhdanskogo vozdushnogo flota, (for Pukhov).
2. Kafedra teoreticheskoy i obshchey elektrotekhniki Kiyevskogo
instituta grazhdanskogo vozdushnogo flota (for Samus').
(Electronic calculating machines)

F~JPPV,-(Ieorgiy-.-Ye,vg-q!llyevich; GLUSHKOV, V.M., akademik, otv.red.;
LABINOVA) U.M., red.3-zd-va; DAM10, Yu.1-1., tekbn. red.
[Calculus of complexes and its application) Kompleksnoe ischi-
slenie i ego primenenie. Kiev) Izd-vo Akad.nauk USSH, 1961. 229 p.
(MIRA 14:12)
1. Akadertiya nauk USSR (for Glushkov).
(Complexes) (Calculus, Operational)

28163
S1144161 /000/009/001 /001
7900 D201/D303
AUTHOR: Pukhov, G.Ye., Corresponding member AS UkrSSR, Doctor of Tech-
- Y@
nica Sciences, 111-ofessor, Head
TITLE: Foundations of the general theory of quasi-analogue systems
d
PERIODICAL: Izvestiya vysshikh uchebnykh zavedeniy. Electromekhanika,l'no.
9, 1961, 3 - 21
TEXT: This article was written from a report, submitted at the ses-
sion of the Physico-Mathematical Section AS UkrSSR on March 29, 1960. The
author describes methods of mathematical representation of some complex sys-
tems by equations which could be set on electronic computers leading to qua-
litative and quantitive solutions. Quasi-analogue systems are defined as
systems, whose representing models have different structures and equations.
A quasi-analogue model of an equation (a) is such that it is a model of dif-
ferent equation (b) which may be partly different from a given equation (a),
but such that it satisfies some determined conditions (the criteria of equi-
valence) and all, or some of the unknowns of the Eq. (b) coincide with the
Cardl/13

28163
S/144/61/000/009/001/001
Founiations of the general ... D201/D303
unknowns of the initial Eq. (a). If the conditions of equivalence are such
that their representation does not require the use of the unknowns resulting
from model (b), then the quasi-analogue model does not differ from an analo-
gue model. These models are called by the author, models of the first kind,
or unbalanced models. If the conditions of the equivalence require the use
of the resulting quantities, then it is necessary to introduce a special pro-
cess, called by the author a balancing model process to satisfy the above
stated conditions, these models are called quasi-analogue models of the sec-
ond kind, or balanced models. Fig. 2 shows the classification of Mathemati-
cal models. Basic problems in a quasi-analogue system are: (a) Deduction
and construction of quasi-analogue models of the first kind, and of quasi-
analogues and balancing systems oi models of the second kind; deducting sigrB
of mathematical equivalence of an initial object and that of a model, and
obtaining criteria of equivalence; (b) detection of and obtaining methods of
study of convergence, establishing the criteria of convergence; (c) working
out ideal systems of quasi-analogue models applicable for the typical ob-
jects. (d) solving the problems of accuracy, automation of balancing, stabi-
lity of the models. The purpose of introducing a quasi-analogue is that by
Card 2113

28163
S/1 44/6 1/000100@11001 /001
Foundations of the general ... D201/D303
introducing known vectors F, G and of a known balancing vector the value
of the unknown vector X could be obtained. The balancing system is used to
transform the vectors Z and H into 4). It is practically convenient that vec-
tor Z obtains, apart from the unknown X, and a vector of auxiliary unknown Y.
also the difference factor F_, whose conversion to zero would mean the full-
fillment of the condition of equivalence. The requirements with respect to
operators are formulated by the author as follows: (a) The operators B and C
should allow for the possibility of construction of a model continuing given
elements. b) The operators B and C should be such that the equation of a qua-
si-analogue model should be found without calculations. This applies also to
operator X and vectort used to determine vector G = K (/,) which, if possib-
le, should be found without long cal-,ulations. c) The operators C and D should
satisfy the conditions of convergence. The author then examines a particular
case the construction of models of equations in which there are no auxiliary
unknowns, and also of equations in which the conditions of equivalence are
not directly obtained. Also examples of quasi-analogue models of the first
kind are given. The difficulty of the problem is in obtaining a quasi-analo-
gue system, The balancing of this system is effected either by hand, or by
simple detecting schemes. Depending on the method of introducing the error
Card 5/

281.63
s/14 61/000/009/001/001
Foundations of the general D201YD303
vector F, and of the balancing vector (Pthere could result different equiva-
lent equations which are suggested by the author. A simple way of construc-
ting mesh quasi-analogues is of the type
AX = F + @ F_
(24)
where @, .s a diLL,-
,,onal matrix, dependent on the structure of the electric cir-
cuit- Modeis of E;q, (24) are called (A-analogues, in Fi@a. 9 the equations
are of the form
@t alixi j- a12X2 t alnxn = F1 + A 1 61
+ a2,xi � a X,, a Xn . F +
22 2n 2
a 1X@ + a a X = F + '2;)
n. n2 nn n n n n F J
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Foundations of the general ... D201/D3O3
In the above scheme the unknowns are represented by the voltages Xi and the
error func tions b,,,, the voltage F_ i. The condition of equivalence is S = O@
The balancing of a scheme shown in Fig. 9 could be solved by Seidel's method.
This method consists of the consecutive equating to zero of the error func-
tions (c, i by the matchi ng of the values of -@ i (f or scheme in Fi g. 9, @@ i = Xi).
This is feasible if the system is convergent, The method consist s of co listruc-
ting an equation linking the vectors I)and 6 by an analysis of a matrix of
coefficients of this equation,
P(P + t = 0
(28)
For CC-analogue models it follows from (24) and (28) and 4)= X, that the ma-
trix P is
po@ 2 A
(30)
When changing in differential equations with ordinary, or with partial deri-
vatives, it may appear that some of the members of the matrix A repeat. Then
the model could be substantially simplified, It would then consist of source
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Foundations of the general D201/D303
of voltage X i , source of current P. and a change over switch. If the matrix
P,x does riot Satisfy Seidel's condition then the balancing of tA-analogues
should be found diffprently. It is possible to apply the relaxation method
cf balancing, The author considers a system shown in Fig. 12, which he calls
a f -analogue, The part. M is represerl4l.ed by
WK @ F + :P (34)
and +.he part D by
DX - tD +
A (35)
where
M A 4 D (36)
and
A , s a d ago nal, ma trJ x, To deduce the condition of convergence Jn the
S-del's in-e@jiod of balancing, eliminating tb,?! ur-knowils frc--.i and (35),
One 6o t,;
(DM-' .. E) DM- F '37)
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Foundations of' the general D20I/D505
As long as D = 1-1 .. Aq the matrix determining the convergence is
Pf- @ - A-1 AM - 1 . (38)
The convergence is quicker the nearer matrix M is to matrix A, A gamma -- ana-
logue model is illustrated by a mesh model of a differential equation
Xiv - 2r2X" + s 4x = q9 (39)
This model permits the solving ofa number of two-dimension diffe *rential,
equations with partial derivatives of a higher than the second order by a
pure machine manner. Such problems are solved on the known mesh electronic
integrators., The author also considered a two member analogue represented by
MX = F 4 (43)
DY = 6, (44)
where M is determined from (36) and the matrix D isselected so that there
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Foundations of the general D20i/D303
are analogue models (43) and (44). The condition of equivalence is at Y = X.
C = x -- Y (45)
which is zero. This model --s called @