DYNAMIC STABILITY OF A CONIC SHELL SUPPORTED ALONG ONE EDGE AND LOADED WITH ACIALLY SYMMETRICAL PRESSURE

• Document Type: CREST
• Collection: General CIA Records
• Document Number (FOIA) /ESDN (CREST): CIA-RDP81-00280R001300170012-8
• Release Decision: RIPPUB
• Original Classification: U
• Document Page Count: 11
• Document Creation Date: December 22, 2016
• Document Release Date: June 8, 2011
• Sequence Number: 12
• Publication Date: October 25, 1956
• Content Type: REPORT

 - STAT
m*11IC STABILITY OF A -CONIC SW.L 3UPlOR'. W ALO*o OR =1 ASH
? LORL'W KITH Afl&TLT 3DQIETRICAL PwSURS
13V..sti Akadsaii Wank MR M. A. Altfpv
,a or the ca~of Sciences V. V. ? Rasaasyss, Nesecu
WWI,, department of tochniosl
sciences, No 10, October 1955,
aaRes 161-165
This paper contains a study of parametric osoilllat.'ons of a
circular conical shell freely supported along one edge. The derived
theoret?cal correlations are united by Us results of tree experiments
performed.
Let us con,ider a thin-walled, circular, conical shell freely
supported along one edge and loaded with external pressure p
&%pending upon time t. We relate the median surface of the shell
to coordinates a and 0 as is shown in ?igure 1.
d (U
;-t (C3'o i) CTQ i + a +i ) ? 0 (l )
ieArwin x is the kinetic ensrxy of mction of the system; U internal
r.%3nt1al energy of the?s~s tea; TT the potential of external forces;
derivative of the general coordinate ai with respect to
t i.e.
Lot us nndsrtake the determination of I, U, and TT appearing
in the Lagrange equations.
For a thin-walled conical shell the internal potential
energy U is expressed by the foraular (1]l
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wherein
U 34(1 = Pr3 l7 (rss t asp + 34M.". + 2 (1 - M) Tel rh as +
a, _ zr. Y,~w( +wowa)+-gafsa
iv'
all f (h' I I, )
^~+-g+y(Ti)+i
a,me __Cwa+ 7 siaa+-}(r + cwa)+3t r abal~
ai r as
or ? r --- 1
Herein u, v, w are the dieplaceaents of the point of
median surface of the shell, respectively, in the direction of
generatrix, the peripheral direction and the direction of the
normal.
K_ JJm'L(+(4r/'+(771'1 d.lr
The potential of external forces we define as
wherein dv in the elea~ent of change in volume circumscribed by the
~.:.~ 12r+w(ei+w r e?aa`a_ PIrA0dS
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Lot us now consider the functions of displacements u, v,
and w by weans of which we have expressed all the quantities
appearing in the equation of Lagrange.
A conic shell freely supported along one edits can undergo
diformation without elongation and shift of median surface. Let
us aseuae that such a deformation is sustained by the shell during
paraaat.ric oscillations. In such a case the functions of displacements
can be determined from the syataa of equations
$a I sty }(mi+-li-(r)s..o
or sp
c,-~ - os?e i slag+-F~eA/+ i ``[ tt) + ( ? -L
.law~s~Q
a. . + as .0
On solving this system approximately with an accuracy to..
a? and taking into account that w . 0 with s - al, we get [?)t
^ak.+at+?t. ?-ri. w.+W,4 vs.
wherein we a On w e,a a
?t~yt aw- Miss a
ti- ? Li.-'t) Cosa -#I tos a am sin a K sLw
w
tt~?i.-.ileosaeT-
w--+F cs - ?tl `.~- sins w[ + eos'w[)
ia'-eos'~N s-stf slasw[
tr,..r,t[a+ a
F -a- ro-w-a is
The appearance of the deformed shell is shown in Figure 2.
Displacements u 0 and wo, corrisrondlnr4 to the symmetrical
forr- of deformation, we will not calcul.,te, s:rre with the selected
funrt ens of diolacanents they have no effect on the c,ances of iotenttal
and kinetic energy of the shell dur.ng parrettic oscillations.
Nft
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Let us calculate K, U, and TT . since it is sufficient to
it follows that up and r2 will appear only in the expression of
the potential of external forces.
On substituting the t..nction of displacements (8) in (3)
and interrating over the entire surface, we Rot
X- (a)' .q, (~a 1 (4'-0.23(4+1)?((I+
After inteRratinn of expression (2) we get
A 4&
U - U, -i- * (
T XI
(,,.A)
.
X.- ~ 1rIa4-2(a'-1)(42-Ml~a)
+ 4(a. - to.~ s)' ~I y) + 4 (4 + 1) (1
FI)
With small angles Cl it is more convenient to calculate U
by expanding Xl to a series by power of (K-:).
4
4, 1 (k --I)'(4i1) + X (4; 1) (1 -0)(a* - rm'a~
]4 4 a
On integrating the expression (L) with rrspect to () within
_ ..
a) S r(+-8irdM
F'9(a' cat
We will assume that the pulsation component of pressure varies
along the v.eneratrlx in proportion to the variation of its static
component, 1.a., we assume the following law of variations of
pressure loadin? the shell:
P.1(') Et + Pe
 wherein A p is a quantity proportional to the aaplitudo of pressure
pulsation.
Let us expand the function f (s) to a Taylor series by
al)
1(i) - Fj