SCIENTIFIC ABSTRACT MAKAROV, B.P. - MAKAROV, F.F.

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R001031510001-9
• Release Decision: RIFPUB
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 4, 2017
• Sequence Number: 1
• Publication Date: December 31, 1967
• Content Type: SCIENCEAB

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soy/17 q- ~ 9- 3 - q/l~ 5
AUTHORS: 8olotin, V. V., Gavrtlov, Yu. V., Makarov, H. P.
Shveyko, Yu. Yu. (Moscow)

TITLE: Non-linear Problems of Stnbilzty of Plane Panels at
High Supersonic Velocities (Nelineynyye zadachi
ustoychivosti ploskikh paneley pti bol'shikh
sverkhzvukovykh skorostyakh)

PERIODICAL: Izvestiya Akademii hank, SSSR, Otdeleniye tekhnicheskikh
hank, Mekhanika I mashinostroyeniye, 1959, Nr 3,
pp 59-6~ (USSR)

ABSTRACT: The paper is a continuation of previous work (Refs 1 and 6).
The Question of the stability of plates nnd shells,
exposed to a ct, trent of compressed gas. has so far been
discussed ir, terms of a linear representation (Refs 1-5).
Fu~ sonic FLow and for moderate supersonic numbers
this hypothesis is apparently completely justified.
However, for lar~er supersonic velocities, aerodynamic
non-linearity becomes very appreciable. As was shown
by Bolotin (Ref 5), solutions different from the
unperturbed ones appear in aeroelastic problems, allowin~
for aerodynamic non-linearity, at velocities below the
Card 1/~ critical value. Among these solutions are some which are

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$0V/179-59-3-9/45
Non-linear Problems of Stability of Plane Panels at High Supersonic
Vel,)cJ ~ies

stable in relation to suf�1ciently small disturbances.
These solutions can be realised if the elastic system
which is sub3ected to ~he sub-critical velocity is
sufficiently irregular. Ali real constructions have some
irregularities (defects of manufacture, deformations
arising ~rom aerodynamic heating, vibrations under the
influence of atmospheric turbulence and other non-
sg~tionar'y factor.q~ etc.). Thus in some cases, the
critical velocity determined by the linear aeroelastic
theory is only a lower limit t~, the critical velocity for
real constructions. In %he present paper, the edges of
the plate are assumed to be simply supported and
elastically restrained ag~ainst axial displacements i the
pressure on the plate is ,given bY:2~
P -- Pup (1 + ~2 1 av ) ~ -1 (1)
where p is the pressure of the unperturb,~d gas, v is
the normal component of surface velocity of the plate,
Card 2/l[ a is the velocity of sound in the unperturbed gas and

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SOV/179-59-.5-9/q5
Non-linear Problems of Stability of Plane panels at Higl, Supersonic
Velocities
x is the polytropy index, The component ol load normal

Card

to the plate is
~ = "Po'0-t2 -

where w is the deflection, ~o is the density and
h the thickness of the plate, � is the damping coefficient
and ~ p is the excess pressure, which can be expressed
in terms of the Mach number and polytropy index by means
of Eq (1). The problem then reduces to the investigation
of the non-linear equation for the deflection of the
plate, which contains q, sub3ect to the bo,]ndary
conditions. One solution is expressed as a double sine
series and is dealt with both by an ap]-roximate
numerical method, and �ith the aid of an electronic
calculating machine. The results of the calculations
for particular cases are shown graphically (Figs q, 5 and
{)), and indicete the existence of flutter in the pane] o
Acknowledgments are ,xpressed to N. I. Chelnokov

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SOV/179-59-3-9/q5
Non-linear Problem, s of ~.,Ollzty of plane Panels at Hxgh Supersonic
Vel. ocitles
and Yu. ~. Shneyde~ of the M~thematical Machlne
Laboratory M~Z , for participe~ing in the calculations.
There are 6 ftgur'es and 9 relerences~ ? of which are
qo',iet and 2 Engllsh.

SU:3MITTED: November 18, 1958

Card

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AUTHORS:

~/'~ 72/6 0/00 a/006/0 ~, 5/0 ~ 8
]3224/])304

Bolotin, V~ V., Doctor of Technical Scier:ces, Professor:
Makarov, B. P., Mishenkov, Go Vo and Scnveyko Yu Yuo
'

TITLE:

An asymptotZc method of ir:vestigating t:.e spectr~mm of
natural frequencies of elastic plates

SOURCE: Raschety na prochnost'; teoreticheskiye ! eksper:men~.
tal'nyye issledovaniya prochnosti mashlz:ostroitel;nykn
konstruktsiy. Sbornik statey. No. 6~ Mo~:ow 1'~60 , /
23 ~-253 '

TEXT: The authors consider the natural vibn'~tions of a rectangular
plate (with the sides a, b) of constant thickness. The general ~o-.
lution of wave equation near the edge x = 0 is looked for in the
form

Card 1/4

W(x, y) = X(x)sin

~K(Y - yo)

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An asymptot:c method ooo
It is deduced that

33397

S/'5 ~2/60/000,./00F~ '0~ '3 ")*:
.)224,.'D304 '

7tx 7~x
x(x) = ClSin ~ + c2cos ~x + C)e

~x ~ ~

Bx � The first lwo terms corres~ond to the asymptotic refte_
sentation for the internal zone; t}-,e third describes the d./r~amlc
edge effecT� Estimation shows that the width of the zone of edge ef-
fect does not exce~ 1/2 of the wavelength. For a plate with ali
edges rigidly fixed,

tg ~ 1 tg
2~x - ~ + 2~2x ' 2~y'

Card 2/4

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3339?
S/5 Y 2/6 0/000/~06/0~ 5,.' 0~
An asymptotic method ..o D224/D304

i.s obtained (~y = 1/~x), ~vhich is reduced to a single trarscenden-
tal equation for ~x' and ~x is computed by successive approxima~:on
the initial value being the asymptotic one 2 = an/bm; t~,e f~nal
~X
is the factordX~= (a/~x)2 + (a/~y)~. The authors
quantity
glve
a

table showing successive stages of computation of o(. for ten
frequencies of a square plate~ and compare all values with Iguti~s
results obtained from a series solution satisfying ail ooundary
ditions (six terms of the series taken)� The largest difference
between the results is 2.53% for m = n = 1. A table of values of
for lb lowest freq~encies of plates with a/b = 0�25 .,d a/b
is also given� The equation for ~x of a pl~te elastically fixed
along all edges is deduced� In this case both ~ and a/'~ must be
x
found by successive approximation; ~ grap~-~ of values of ~Y as
function of ]~ = 27~'D/ac (D being ~he cylindrlcal rzgzdity of the
plate~ c the rigidity factor for the edge) for 10 types of vibr~-
tion is given� The case of an axially compressed plate is treat,:~,
Card 3/4

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Arz asymptotic method 8./'572/60/000,/0..~,~,/0~. 5
' ' ' ~22~/D304
in the same way� Pour ca,es ~re considered next, In which some sz-
de~ are hinged and other ~ide~ rigidly ~lxed. Values off. computed
for the~e ca~e~ for a square plate a~e tabul:-~ted and compared w~th
those obtained by Ritz's method. The authors remark that some for-
mulae for pzlncipal frequencies by Ritz's method, given in other
Publications, also in two reference manuals~ contain errors. Eq~a-
tion for ~x of an .rrb. tropic plate is also derived an~ a table of
~ for a square plate is given. There are 9 figures, 6 tables and
14 references: 1~ Soviet-bloc md 3 non--Soviet-bloc. The referenc~
to the English-language publication reads as follows: F. Priede.-
richs~ Asymptotic pheaomena in mathematical physics. Bull ~er~ ,
Math. Soc. 6~ no. 6, 1955. � c

Card 4/4

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. / ,24/62/ooo/o12/oo6/o o9
/'!" ":""J D2~4/D303

~'~J3'i~{OR: 1.; akarov ~ B .z'.
TITLE: Nonlinear flutte~ of a plate cl~,l)cd alc, r~ its edge

i'ERIODiC.~: ![eferati~yy zhu~al, Iiel:h~it;a, no. 12, 1962, 27,
abstract 122159 ('l'r. lio~eren~sii po teorii plastin
i obolochek, 1960, i,an (Ref. Q), a theore%ical curve
ratio q~/q~ as a func%lon of the paran~eter u is constructed.
This parameter is defined by; _

-, h o 2
U = >o ~ n m
-..'nt-~-_ . 0:3 o is the aml~litude of Initial deflection; n the
r.~. ;.r of waves in circumfel'cntial direction; m = (s/~ x the
raLlo of the length of a clrcun~ferential half wave to the
of a longitudinal one. Finally, ~nowing the exper~a,entai
dlatribution of the critical stress ~nd ~he theoret lcal
bct'.~een the stress and the I~araineter u, an experl;:~enta 1
.~ oLrlbution of u is de%erlninea using the usual probability
n.cti~ods. Ehe results of t~ese calculations are l, re~el:ted
i.'l._].-} in the form of a histogram, on which, for comparison, the

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Al~pl�cation of statistical method... S/179/62/000/001/022/0a7
Etl~/E155
normal distribution curve with the mean, m = U. i, and ~he
standard deviation, 0 = 0.06, is plotted. Tho ~'uthor ~tr,'~,.~es
%nat tho experimental data available %o him are insul'; ,, ~,nt J (,r
dz'awir'~ any firm conclusions, and Lb.it his p~l,cz' "~l~(,u 1,1 ),,~
consi~e~_d only us a first attempt Lo ustlmate %h,t cln,,'~, (,'r ,~
the d~.~tribution of critical stresses and puram,'ters of ]~,~t s;,~
imperiections on the basis of exper/mental data"
There are % fiEures.
SUBb;ITTI~D: October 7, 1961

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BO!Y)TIN, V.V., doktor teknn.nnuk, prof.; ~..M_A~AROV, a.P,, kand.te~hn.nauk;
}O?RANOV, B.A., inzn.

Strength and rigidity ~f lntern~l transf~,rme: ~[nd.l ~r.,
Elektriche s%vo no.A: 5A-~2 Ap ' "~ � "H2 ~' ~ 1~'..

1. Moskovskiy energeticbe.~ki� in.~tltut.

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KURANOV, B.A.~ aspirant; M~.KAROV, B.l., ~.a3,a. te.