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On the dynamic stability of thin elastic shells filled with a liquid
ON TEE DYNAMIC STABILITY OF TKIN ELASTIC SHELLS FILLED WITH A LIQUID (0 DINAIUICHESKOI OBOLOCHEK PMM Vo1.24,
B. N.
USTOICHIVOSTI TONKIKH UPRUGIKH NAPONENNYKH ZMIDKOST'YU) No.5,
1960,
BUBLIK and V. I. (Kiev)
(Received
25
pp.
94i-946
MERKULOV
November
1959)
We shall study in the following the dynamic stability of a thin elastic shell filled partly or completely with an ideal incompressible liquid. Use will be made of a paper by Moiseev [ 1 1 which deals with free vibrations of an elastic beem containing a liquid, as well as of [ 2 1 and [ 31, devoted to the study of dynamic and static stability of elastic systems without a liquid. 1. The problem indicated equation
reduces
to the solution
of the variational
1
8
s 10
(T”-A”--U”)dt
~0
where T”..and U”..are the kinetic and the potential of the perturbed system, so that
(1.1)
energies,
respectively,
while the quantity A” represents the work of a certain reduced loading, arising as a result of perturbations of the system, done in the displacements of such perturbations. The determination of the reduced loading is achieved in the same manner as in [ 2 1. In such cases when, in the determination of these loadings, the inertial terms can be neglected. or when the initial state (the state which is to be investigated for stability) of the shell is approximately a membrane state, they will be
1423
B.N.
1424
F~ = p+ C+ W’T2°) -
T1°
Bublik
6
(E, P) + 2 F,
F,
=
&[g
(~zQTl'1
-
T2"
and V.I.
=
@zpS”) + So &
(ezQ) -
qP (el
+
e,,]
+ Taoxz
TIM
(8zQ) + G
-&
Merkulou
(1.4)
(81Qs") + So-$ (61~) -9,
(~1 +
es) ]
and then
A” = ;
[F,u
+ Fpv + F,w]d6
(1.5)
c
Here and in the following we shall denote by: 2 the middle surface of the shell: a, /3 the curvilinear orthogonal coordinates in the middle surP, Q the coefficients of the face; n a normal to the middle surface; first quadratic form of the middle surface; II, V, 1 the displacement components of a perturbation of the shell in the directions of a, p, n, reof unit surface area of the shell and spectively; us, p mass densities unit volume of the liquid, respectively; c 1, c2, q~,, K~, r relative deformations of the shell, expressible In terms of u, (I, v by means of well-known formulas of the linear theory of shells; T1, Tx, S, Ml, M2, If stress resultants and couples of the shell, expressible in terms of the relative deformations by means of Hooke’s law; Tlo, Tzo, 9 stress resultants of the non-perturbed shell, which characterize the initial membrane state of the shell: pa, q, q, external surface loadings, acting on the shell, possibly functions of time; a the acceleration of the translational motion of the system; S the free surface of the liquid in 4 velocity the state of rest; V the volume occupied by the liquid; potential of ,the liquid in that domain; z( ‘) the wetted surface of the shell;xl the part of the boundary of V where the derivative dp/ ah is known; 8, the part of the boundary of V where the function q%is known; G Green’s function of Neumann-Dlrichlet's problem for Laplace’ 8 equation in the domain V. The potential manner:
C$can be expressed
The solution of Equation differential equations
(1.1)
in terms of G in the following
reduces
i-9 h(u)
+
&2(v)
+
.513(4
+
- &
to the solution
F,-m,,
a%L at2 =O I
of the four
Dynamic
stability
of
thin
elastic
1 -v”[* J&(u)
+ -522(V) + I;zs(w) +
- Bh
L
p-
shells
1
l32V 'jlo_ =
azw
b(u)
+Ls2(V)
+&3(w)
mo~-Q~
+
1425
(1.7)
1
acp
Acp=O with boundary conditions for the domain V. fore we do not them explicitly.
0
=o
conditions corresponding to fixed edges of the shell, the velocity potential of the liquid on the boundary The edges of the shell can be fixed in various ways, formulate here the boundary conditions corresponding The conditions for $ are
aq ---
an -
a-!$=0
on the free
aw
on the wetted middle surface of the shell
at
surface
and of thereto
z = 0
(1.8) (1.9
The operators Lll, . . . . LS3 appearing in Equations (1.7) are the lefthand members of the equations of the theory of shells [ 4.5 1. 2. It is not difficult to introduce E, N and a vector X(u, v, w, 4). the form
L,
into the discussion such operators so that Equations (1.7) appear in
I,
LX+MX+E,,,+N
3x
ax at=O
The operators L, M, E, N satisfy the conditions of existence and uniqueness of the generalized solution in accordance with Theorem 3 of 16 I. an application of the theory developed above we consider In the following the question of dynamic stability of a circular cylindrical shell filled with a liquid and with hinged edges. 3.
As
Let us introduce a system of cylindrical coordinates r, z, 8 oriented In the usual way. Assume the cylinder under consideration of radius R to be situated between the planes t = 0 and z = 1. The inner space of the cylinder Is partly filled with a liquid of density p up to a level z = Z1. We choose the curvilinear coordinates a and @ In such a manner as to have B = 8, Q = s/R, where 6 is the angular coordinate of the cylindrical system. Assume the cylindrical shell distributed radial loading q,(a,
considered to be under the action of a t). This can be, for example, a hsdro-
1426
B.N.
static loading. The shell distributed load qa(a, t)
Bublik
is,
and
V.I.
Merkulov
in addition,
acted
and by a longitudinal
upon by a longitudinally compressive
force
F(t).
For the case under consideration the coefficients of the first quadratic form will be P = Q = R, while the curvatures are K, = 0, K* = l/R. We assume that the shell characterized by the stress
is initially resultants
Integrating the equations state of stress we obtain
in a membrane state TIO,
of stress,
T2”.
of the cylindrical
shell
in the membrane
a
T1° = R
s
qa (a, t) da -
&
F (t),
T2’ = Ilq,(a,
tj
(3.1)
0
The relative
deformations
of the cylindrical
The components of the reduced loading which lead to
shell
will
are determined
be
by Formulas (1.4).
1 F,
= ~2
1 Fp =p(T2-T1=&q,
(T,” -
Tao)
@u
we now turn to the determination motion of the liquid.
Pa=-&{T;g
of the velocity
+
T;(g+
potential
w>}
(3.3)
4 of the
Relaxing the boundary conditions at the section z = 0 and neglecting the wave motion on the free surface, we shall assume that at the sections z = 0 and z = I, we know the additional pressure p produced by the perturbation of the system under consideration. It is obvious that such an assumption is justified in the case of long cylinders, because in this case the energy of the wave motion is small as campared with the total energy of the system and, besides, long cylinders lose stability in a non-axisymmetrical deformation, when
which means absence of expulsion of fluid. i.e. P = 0. ConsequentlY, the function + is known at the sections z = 0 and z = II; we consider these sections as the domain I;. Cn the lateral surface we know the derivative
Dynamic
a+/& our
= &/at.
stability
therefore
of
we choose
thin
elastic
the
wetted
4 from Equation
(1.6).
shells
surface
1427
of
the
shell
as
domain xl. In order
function dG/d,
to determine
for
the
we write
down Green’s
cylinder
0 < a < ll/R, r < R, with the and G = 0 when a = 0 or a = Z1/R:
= 0 when r = R, co
G= +
condition
that
00
x
x’sin
p=1
&a)
sin &u’)
g,, (r,r’) cos (p -
p’) 4
(3.4)
q=o
where Clp’F
A prime should
1, (ppr’/ R)
PZR
gp, cry f) = at the
sign
of
I -+
PU
aau
The tangential and those
of
1+ v
aav
half
usually
neglected
normal
components use
is
reduced
the
@(a,
first
this @,
C9W
n,,d2 u/a t2,
- 1, (pp +) term of
of
system
l-v
K,’ hp)]
order
r,,d2dd
t2 of
12
1
zero
will
and if to
the old
be fulfilled
determination
(V 2 +i)~V~v~0-_(1-v)~2(~-~)vm+~
=O (3.5)
1
=0
inertia
forces
Fa and F
are B’ with the
respectively.
connected
for
we obtain
I
in comparison
we introduce variables
a50 \ 630 -hap4 J + 3Z@-v a30 as0 T7*0-(2+v)asap +aps+
serve
the
the components
shells
(1.7),
F,--m& [ i LFB-mOg
i-v Eh
simplification
t),
the
Eh
namely
t2 and F,.
two equations will
of
+v&+
loading,
r0a2v/d
made of
equation
a+
in th’e theory
t’=2G& the
equations
i-va2v aw -2 aa2 +,P+
components
the
known function tions
third
-?--a%@
1 + v aau --F-SZ@+f+
third
sum means that
4 from the
af2apP’
then
[Ip’ (~~1 K, (rp +)
be taken.
Eliminating
If
I,’ @,)
a new unrela-
by the
as0 53 w = y7Ty0 identically,
while
of @ i
-
9 a40
aa4+
the
1428
B.N.
Bublik
and
V.I.
Merkulov
(3.7)
In order
to satisfy PW
m = 0,
0,
W.=
the boundary conditions
the function
I
T1 = 0
v = 0,
(3.8)
for a=0andu=x
we prescribe
for
Satisfying differential series
these conditions we try to find the solution of the iutWroequation (3.7) by the method of Galerkin in the form of a
cI, (a, 8, t) =
5i
@ the boundary conditions
f,,(t)
sin &,a cos 43
(1, = y)
13.10)
n=o m=o
Substitution 00
of
(3.10)
into Equation
(3. ‘7) gives
03
H,zo{[
(A,” + n2 + 1)s (a,”
+ yhm2]f
+
ny +
sin h,u cos n@ + 7
mn
- % ITI”&
f
Tea (nz-
(i -
v)
a,=(a*2 -
(Am2 + na)
sin &a’ cos n@‘da’
\‘,
Let us consider
in more detail
x cos g (p -
j&t sin h,a cos nfi sin &,,a cos nf3
t)] (hma + n”)zfm,
G (&,,2 + na) f,i
the last
n2) (jLgn2 -t- aa)+
>
= 0
term of this
p’) cos np’da’da
x fm @sin pLpacos nB
-I-
(3.11)
equation.
=
(3.12)
Dynaric
stability
of
Substituting this expression coefficients of co6 n/3 to zero,
thin
into
elastic
Equation
we obtain
shells
(3.11)
1429
and equating
the
the system of equations
co
(Lx” + na+ II2 (a,,’ + n’)’ + (1 - V)Am2(A,’ - n4) + ‘! am41 f,, 2 1r $ ITI~J.,’ + Tao(n” - I)] (A,’ + n$)*f,, sin ~,,,a + F (A,,%+ n2)f,, sin A, o + sin &a
-
rn=”
-
f
5
(-l)P~gp,r(R,
R)%‘T:Tp (n =
(3.131,
sin~f,~.ginp,u}=O
P=O
3, . Y.,
I,;,
Multiplying the last equation by sin Xia and integrating limits 0 and a = 1/R. we obtain for determination of f,,,(t) infinite coupled system of equations:
WI=0
Here, the notation J.(l) rn = &
$
J mm @!f mn = 0
m=o
C
between the the following
(n, i = 1,2,.
. .)
(3.14)
is introduced
(h: + 11.~ + 1)2 (AC + n2) + (1 -v)
AC (Q - n4) + ‘$
&4]
l/R
J
(1) _
mni -
+
$
R 1.z (h,s + .z)a
D
wn
(II?' - 1) (Am2 +
s 0 l/R
nf) \
TI” (a, t)
sin &,a sin @da
+
Tz” (CL,t) sin h,u sindiuda
1
/ mni = i
p=o
(-I)p+i
v
gpn (RIR) (a JFiT m
) sin m~zl sin F
7’ 21 P
P
ia
The applicability of the method of Galerkin to the solution of the problem under consideration proves the convergence of the system of equations (3.14). This leads to the conclusion that an approximate solution of this infinite system can be obtained by using a truncated finite system of equations (3.14) with n, i = 1, . . . , IV. As a result we shall obtain a system of Hill’s equations, and an investigation of this system will permit us to answer the question concerning the natural frequencies and critical forces for the system, consisting of shell and liquid, and to establish the region of dynamic stability in the space of the system parameters n/R, 1,/R, 1/R, a,, p for various external loadings.
1430
B.N.
Bublik
and
V.I.
Yerkulov
4. Let us consider in greater detail the case when the liquid fills the entire inner space of the cylindrical shell. This corresponds to the case I = I,. In the infinite sum I ani there remains the term p = 1 and, further, I . = 0 for all numbers II f i. For II = i we obtain mna Iini = Ini = qf
by
Furthermore. we replace the mean values TlO*(t) = :
Then the coefficient: R = i they become J,(z) 2n
=
J,(2) mi
=
R 2o1
gin
the stress
resultants
vanish
J!,ii
Ai2 (3,:
+
n2)
‘2 + [J,l$
This system is conveniently
-
(4.1)
Tlo (a.
t)
and TzO(a,
1/n R . T,“* = _ 1 i Tzo (a, t) da
\ Tl” (a, t) da, for
all
Tl”* (t) + &
By virtue of these simplifications reduces to a system of ordinary Hill (Zino + Zin)
(R, R) (hi2 + P)
numb:rs I f
(h2 -
coupled
for
in the following
(4.4)
form:
(n, i = 1, 2, . . .)
fin(t)=0
(4.3)
system (3.14)
(i, II = 1, 2, . . .)
J,:) (t)] fi, (t) = 0
d~++_xp-T~)
(4.2)
1) (AC + hz) Tao* (t)
the infinite equations
written
i. while
t)
(4.5)
The quantities [(inR/Zja + n2 + 112 D O’2rl’ = R4 [m. + 4npRg,, (R,R)] D (1 -v) +
+
(iaxR / I)%[(inR / 1)4 - n4] + (in;R / Z)4(1 - 9) / la R4 [m. -I- 4nRpg,, (R,R)I I(inR IV -I- nala
(4.6)
appearing in (4.5) are of the nature of natural frequencies of the shell; the quantity 4npRgin(R, R) represents the added mass due to the presence of the liquid in the shell. The quantities T1* =
ain2 Imo -I- 4aRQgin(R, (inR 1 l)2
41
Tat = - %’ n= -
1
[m, + 4fiQRgi,j(R, R)l
represent the critical longitudinal and normal loadings; their Physical or Tao* exceed those meaning is that if the stress resultants TP* values the solution of Equations (4.5) will increase exponentially, which corresponds to the loss of stability of the shell. The analysis
of the dynamic stability
of the shell
can be carried
out
Dynanic stability
of
thin elastic
1431
shells
with the aid of a diagram of stability of Hill’s equation (4.5). In this case it will be possible to indicate for any value of shell parameters whether the latter will be stable under the action of given forces. It must be stated that the presence of liquid masses causes considerable decrease of the natural frequencies for the first harmonics of the bending mode. In some examples the ratio of decrease amounts to several hundred. It is for loss liquid. liquid [ possible
also essential that the critical values of the stress resultants of static stability are not influenced by the presence of a It is, in analogy to the case of vibrations of a beam with a 1 I , impossible to indicate an equivalent shell, although it is to indicate the equivalent mass of shell for each bending mode.
The authors express of the problem and for
their gratitude to N.N. Moiseev for his frequent valuable advice.
the statement
BIBLIOGRAPHY 1.
kolebanii uprugikh tel. imeiushchikh zhidkie Moiseev. N. N., K teorii polosti (On the theory of vibrations of elastic bodies having cavities filled with liquid). PMM Vol. 23. NO. 5. 1959.
2.
Bolotin,
V. V.,
Stability
3.
Novozhilov. the
of
Dinaricheskaia Elastic
V. V.,
Nonlinear
Osnovy
Theory
4.
Vlasov, V. Z. , Obshchaia Gostekhizdat. 1949.
5.
Gol’ denveizer, Elastic
6.
Thin
A. L., Shells).
uprogikh GITTL, 1956.
astoichivost’
Systems). nelineinoi of
teorii
obolochek
Teoriia aprugikh GITTL, 1953.
(General
t*nkikh
(Dynamic
(Foundations
Gostekhizdat,
Elasticity). teoriia
uprugosti
sister
of
1947. Theory
oboloehek
of
(Theory
Shells).
of
Vishik, M.I., Smeshannye krayevye zadachi dlia sistem uravnenii, soderzhashchikh vtorniu proizvodnuiu PO vremeni i priblizhennyi metod ikh resheniia (Mixed boundary-value problems for systems of equations, containing the second derivative with respect to time and an approximate method for their solution). Dokl. Akad. Nauk, 19, Vol. 100, No. 3.
Translated
by
1.M.
B. N.
USTOICHIVOSTI TONKIKH UPRUGIKH NAPONENNYKH ZMIDKOST'YU) No.5,
1960,
BUBLIK and V. I. (Kiev)
(Received
25
pp.
94i-946
MERKULOV
November
1959)
We shall study in the following the dynamic stability of a thin elastic shell filled partly or completely with an ideal incompressible liquid. Use will be made of a paper by Moiseev [ 1 1 which deals with free vibrations of an elastic beem containing a liquid, as well as of [ 2 1 and [ 31, devoted to the study of dynamic and static stability of elastic systems without a liquid. 1. The problem indicated equation
reduces
to the solution
of the variational
1
8
s 10
(T”-A”--U”)dt
~0
where T”..and U”..are the kinetic and the potential of the perturbed system, so that
(1.1)
energies,
respectively,
while the quantity A” represents the work of a certain reduced loading, arising as a result of perturbations of the system, done in the displacements of such perturbations. The determination of the reduced loading is achieved in the same manner as in [ 2 1. In such cases when, in the determination of these loadings, the inertial terms can be neglected. or when the initial state (the state which is to be investigated for stability) of the shell is approximately a membrane state, they will be
1423
B.N.
1424
F~ = p+ C+ W’T2°) -
T1°
Bublik
6
(E, P) + 2 F,
F,
=
&[g
(~zQTl'1
-
T2"
and V.I.
=
@zpS”) + So &
(ezQ) -
qP (el
+
e,,]
+ Taoxz
TIM
(8zQ) + G
-&
Merkulou
(1.4)
(81Qs") + So-$ (61~) -9,
(~1 +
es) ]
and then
A” = ;
[F,u
+ Fpv + F,w]d6
(1.5)
c
Here and in the following we shall denote by: 2 the middle surface of the shell: a, /3 the curvilinear orthogonal coordinates in the middle surP, Q the coefficients of the face; n a normal to the middle surface; first quadratic form of the middle surface; II, V, 1 the displacement components of a perturbation of the shell in the directions of a, p, n, reof unit surface area of the shell and spectively; us, p mass densities unit volume of the liquid, respectively; c 1, c2, q~,, K~, r relative deformations of the shell, expressible In terms of u, (I, v by means of well-known formulas of the linear theory of shells; T1, Tx, S, Ml, M2, If stress resultants and couples of the shell, expressible in terms of the relative deformations by means of Hooke’s law; Tlo, Tzo, 9 stress resultants of the non-perturbed shell, which characterize the initial membrane state of the shell: pa, q, q, external surface loadings, acting on the shell, possibly functions of time; a the acceleration of the translational motion of the system; S the free surface of the liquid in 4 velocity the state of rest; V the volume occupied by the liquid; potential of ,the liquid in that domain; z( ‘) the wetted surface of the shell;xl the part of the boundary of V where the derivative dp/ ah is known; 8, the part of the boundary of V where the function q%is known; G Green’s function of Neumann-Dlrichlet's problem for Laplace’ 8 equation in the domain V. The potential manner:
C$can be expressed
The solution of Equation differential equations
(1.1)
in terms of G in the following
reduces
i-9 h(u)
+
&2(v)
+
.513(4
+
- &
to the solution
F,-m,,
a%L at2 =O I
of the four
Dynamic
stability
of
thin
elastic
1 -v”[* J&(u)
+ -522(V) + I;zs(w) +
- Bh
L
p-
shells
1
l32V 'jlo_ =
azw
b(u)
+Ls2(V)
+&3(w)
mo~-Q~
+
1425
(1.7)
1
acp
Acp=O with boundary conditions for the domain V. fore we do not them explicitly.
0
=o
conditions corresponding to fixed edges of the shell, the velocity potential of the liquid on the boundary The edges of the shell can be fixed in various ways, formulate here the boundary conditions corresponding The conditions for $ are
aq ---
an -
a-!$=0
on the free
aw
on the wetted middle surface of the shell
at
surface
and of thereto
z = 0
(1.8) (1.9
The operators Lll, . . . . LS3 appearing in Equations (1.7) are the lefthand members of the equations of the theory of shells [ 4.5 1. 2. It is not difficult to introduce E, N and a vector X(u, v, w, 4). the form
L,
into the discussion such operators so that Equations (1.7) appear in
I,
LX+MX+E,,,+N
3x
ax at=O
The operators L, M, E, N satisfy the conditions of existence and uniqueness of the generalized solution in accordance with Theorem 3 of 16 I. an application of the theory developed above we consider In the following the question of dynamic stability of a circular cylindrical shell filled with a liquid and with hinged edges. 3.
As
Let us introduce a system of cylindrical coordinates r, z, 8 oriented In the usual way. Assume the cylinder under consideration of radius R to be situated between the planes t = 0 and z = 1. The inner space of the cylinder Is partly filled with a liquid of density p up to a level z = Z1. We choose the curvilinear coordinates a and @ In such a manner as to have B = 8, Q = s/R, where 6 is the angular coordinate of the cylindrical system. Assume the cylindrical shell distributed radial loading q,(a,
considered to be under the action of a t). This can be, for example, a hsdro-
1426
B.N.
static loading. The shell distributed load qa(a, t)
Bublik
is,
and
V.I.
Merkulov
in addition,
acted
and by a longitudinal
upon by a longitudinally compressive
force
F(t).
For the case under consideration the coefficients of the first quadratic form will be P = Q = R, while the curvatures are K, = 0, K* = l/R. We assume that the shell characterized by the stress
is initially resultants
Integrating the equations state of stress we obtain
in a membrane state TIO,
of stress,
T2”.
of the cylindrical
shell
in the membrane
a
T1° = R
s
qa (a, t) da -
&
F (t),
T2’ = Ilq,(a,
tj
(3.1)
0
The relative
deformations
of the cylindrical
The components of the reduced loading which lead to
shell
will
are determined
be
by Formulas (1.4).
1 F,
= ~2
1 Fp =p(T2-T1=&q,
(T,” -
Tao)
@u
we now turn to the determination motion of the liquid.
Pa=-&{T;g
of the velocity
+
T;(g+
potential
w>}
(3.3)
4 of the
Relaxing the boundary conditions at the section z = 0 and neglecting the wave motion on the free surface, we shall assume that at the sections z = 0 and z = I, we know the additional pressure p produced by the perturbation of the system under consideration. It is obvious that such an assumption is justified in the case of long cylinders, because in this case the energy of the wave motion is small as campared with the total energy of the system and, besides, long cylinders lose stability in a non-axisymmetrical deformation, when
which means absence of expulsion of fluid. i.e. P = 0. ConsequentlY, the function + is known at the sections z = 0 and z = II; we consider these sections as the domain I;. Cn the lateral surface we know the derivative
Dynamic
a+/& our
= &/at.
stability
therefore
of
we choose
thin
elastic
the
wetted
4 from Equation
(1.6).
shells
surface
1427
of
the
shell
as
domain xl. In order
function dG/d,
to determine
for
the
we write
down Green’s
cylinder
0 < a < ll/R, r < R, with the and G = 0 when a = 0 or a = Z1/R:
= 0 when r = R, co
G= +
condition
that
00
x
x’sin
p=1
&a)
sin &u’)
g,, (r,r’) cos (p -
p’) 4
(3.4)
q=o
where Clp’F
A prime should
1, (ppr’/ R)
PZR
gp, cry f) = at the
sign
of
I -+
PU
aau
The tangential and those
of
1+ v
aav
half
usually
neglected
normal
components use
is
reduced
the
@(a,
first
this @,
C9W
n,,d2 u/a t2,
- 1, (pp +) term of
of
system
l-v
K,’ hp)]
order
r,,d2dd
t2 of
12
1
zero
will
and if to
the old
be fulfilled
determination
(V 2 +i)~V~v~0-_(1-v)~2(~-~)vm+~
=O (3.5)
1
=0
inertia
forces
Fa and F
are B’ with the
respectively.
connected
for
we obtain
I
in comparison
we introduce variables
a50 \ 630 -hap4 J + 3Z@-v a30 as0 T7*0-(2+v)asap +aps+
serve
the
the components
shells
(1.7),
F,--m& [ i LFB-mOg
i-v Eh
simplification
t),
the
Eh
namely
t2 and F,.
two equations will
of
+v&+
loading,
r0a2v/d
made of
equation
a+
in th’e theory
t’=2G& the
equations
i-va2v aw -2 aa2 +,P+
components
the
known function tions
third
-?--a%@
1 + v aau --F-SZ@+f+
third
sum means that
4 from the
af2apP’
then
[Ip’ (~~1 K, (rp +)
be taken.
Eliminating
If
I,’ @,)
a new unrela-
by the
as0 53 w = y7Ty0 identically,
while
of @ i
-
9 a40
aa4+
the
1428
B.N.
Bublik
and
V.I.
Merkulov
(3.7)
In order
to satisfy PW
m = 0,
0,
W.=
the boundary conditions
the function
I
T1 = 0
v = 0,
(3.8)
for a=0andu=x
we prescribe
for
Satisfying differential series
these conditions we try to find the solution of the iutWroequation (3.7) by the method of Galerkin in the form of a
cI, (a, 8, t) =
5i
@ the boundary conditions
f,,(t)
sin &,a cos 43
(1, = y)
13.10)
n=o m=o
Substitution 00
of
(3.10)
into Equation
(3. ‘7) gives
03
H,zo{[
(A,” + n2 + 1)s (a,”
+ yhm2]f
+
ny +
sin h,u cos n@ + 7
mn
- % ITI”&
f
Tea (nz-
(i -
v)
a,=(a*2 -
(Am2 + na)
sin &a’ cos n@‘da’
\‘,
Let us consider
in more detail
x cos g (p -
j&t sin h,a cos nfi sin &,,a cos nf3
t)] (hma + n”)zfm,
G (&,,2 + na) f,i
the last
n2) (jLgn2 -t- aa)+
>
= 0
term of this
p’) cos np’da’da
x fm @sin pLpacos nB
-I-
(3.11)
equation.
=
(3.12)
Dynaric
stability
of
Substituting this expression coefficients of co6 n/3 to zero,
thin
into
elastic
Equation
we obtain
shells
(3.11)
1429
and equating
the
the system of equations
co
(Lx” + na+ II2 (a,,’ + n’)’ + (1 - V)Am2(A,’ - n4) + ‘! am41 f,, 2 1r $ ITI~J.,’ + Tao(n” - I)] (A,’ + n$)*f,, sin ~,,,a + F (A,,%+ n2)f,, sin A, o + sin &a
-
rn=”
-
f
5
(-l)P~gp,r(R,
R)%‘T:Tp (n =
(3.131,
sin~f,~.ginp,u}=O
P=O
3, . Y.,
I,;,
Multiplying the last equation by sin Xia and integrating limits 0 and a = 1/R. we obtain for determination of f,,,(t) infinite coupled system of equations:
WI=0
Here, the notation J.(l) rn = &
$
J mm @!f mn = 0
m=o
C
between the the following
(n, i = 1,2,.
. .)
(3.14)
is introduced
(h: + 11.~ + 1)2 (AC + n2) + (1 -v)
AC (Q - n4) + ‘$
&4]
l/R
J
(1) _
mni -
+
$
R 1.z (h,s + .z)a
D
wn
(II?' - 1) (Am2 +
s 0 l/R
nf) \
TI” (a, t)
sin &,a sin @da
+
Tz” (CL,t) sin h,u sindiuda
1
/ mni = i
p=o
(-I)p+i
v
gpn (RIR) (a JFiT m
) sin m~zl sin F
7’ 21 P
P
ia
The applicability of the method of Galerkin to the solution of the problem under consideration proves the convergence of the system of equations (3.14). This leads to the conclusion that an approximate solution of this infinite system can be obtained by using a truncated finite system of equations (3.14) with n, i = 1, . . . , IV. As a result we shall obtain a system of Hill’s equations, and an investigation of this system will permit us to answer the question concerning the natural frequencies and critical forces for the system, consisting of shell and liquid, and to establish the region of dynamic stability in the space of the system parameters n/R, 1,/R, 1/R, a,, p for various external loadings.
1430
B.N.
Bublik
and
V.I.
Yerkulov
4. Let us consider in greater detail the case when the liquid fills the entire inner space of the cylindrical shell. This corresponds to the case I = I,. In the infinite sum I ani there remains the term p = 1 and, further, I . = 0 for all numbers II f i. For II = i we obtain mna Iini = Ini = qf
by
Furthermore. we replace the mean values TlO*(t) = :
Then the coefficient: R = i they become J,(z) 2n
=
J,(2) mi
=
R 2o1
gin
the stress
resultants
vanish
J!,ii
Ai2 (3,:
+
n2)
‘2 + [J,l$
This system is conveniently
-
(4.1)
Tlo (a.
t)
and TzO(a,
1/n R . T,“* = _ 1 i Tzo (a, t) da
\ Tl” (a, t) da, for
all
Tl”* (t) + &
By virtue of these simplifications reduces to a system of ordinary Hill (Zino + Zin)
(R, R) (hi2 + P)
numb:rs I f
(h2 -
coupled
for
in the following
(4.4)
form:
(n, i = 1, 2, . . .)
fin(t)=0
(4.3)
system (3.14)
(i, II = 1, 2, . . .)
J,:) (t)] fi, (t) = 0
d~++_xp-T~)
(4.2)
1) (AC + hz) Tao* (t)
the infinite equations
written
i. while
t)
(4.5)
The quantities [(inR/Zja + n2 + 112 D O’2rl’ = R4 [m. + 4npRg,, (R,R)] D (1 -v) +
+
(iaxR / I)%[(inR / 1)4 - n4] + (in;R / Z)4(1 - 9) / la R4 [m. -I- 4nRpg,, (R,R)I I(inR IV -I- nala
(4.6)
appearing in (4.5) are of the nature of natural frequencies of the shell; the quantity 4npRgin(R, R) represents the added mass due to the presence of the liquid in the shell. The quantities T1* =
ain2 Imo -I- 4aRQgin(R, (inR 1 l)2
41
Tat = - %’ n= -
1
[m, + 4fiQRgi,j(R, R)l
represent the critical longitudinal and normal loadings; their Physical or Tao* exceed those meaning is that if the stress resultants TP* values the solution of Equations (4.5) will increase exponentially, which corresponds to the loss of stability of the shell. The analysis
of the dynamic stability
of the shell
can be carried
out
Dynanic stability
of
thin elastic
1431
shells
with the aid of a diagram of stability of Hill’s equation (4.5). In this case it will be possible to indicate for any value of shell parameters whether the latter will be stable under the action of given forces. It must be stated that the presence of liquid masses causes considerable decrease of the natural frequencies for the first harmonics of the bending mode. In some examples the ratio of decrease amounts to several hundred. It is for loss liquid. liquid [ possible
also essential that the critical values of the stress resultants of static stability are not influenced by the presence of a It is, in analogy to the case of vibrations of a beam with a 1 I , impossible to indicate an equivalent shell, although it is to indicate the equivalent mass of shell for each bending mode.
The authors express of the problem and for
their gratitude to N.N. Moiseev for his frequent valuable advice.
the statement
BIBLIOGRAPHY 1.
kolebanii uprugikh tel. imeiushchikh zhidkie Moiseev. N. N., K teorii polosti (On the theory of vibrations of elastic bodies having cavities filled with liquid). PMM Vol. 23. NO. 5. 1959.
2.
Bolotin,
V. V.,
Stability
3.
Novozhilov. the
of
Dinaricheskaia Elastic
V. V.,
Nonlinear
Osnovy
Theory
4.
Vlasov, V. Z. , Obshchaia Gostekhizdat. 1949.
5.
Gol’ denveizer, Elastic
6.
Thin
A. L., Shells).
uprogikh GITTL, 1956.
astoichivost’
Systems). nelineinoi of
teorii
obolochek
Teoriia aprugikh GITTL, 1953.
(General
t*nkikh
(Dynamic
(Foundations
Gostekhizdat,
Elasticity). teoriia
uprugosti
sister
of
1947. Theory
oboloehek
of
(Theory
Shells).
of
Vishik, M.I., Smeshannye krayevye zadachi dlia sistem uravnenii, soderzhashchikh vtorniu proizvodnuiu PO vremeni i priblizhennyi metod ikh resheniia (Mixed boundary-value problems for systems of equations, containing the second derivative with respect to time and an approximate method for their solution). Dokl. Akad. Nauk, 19, Vol. 100, No. 3.
Translated
by
1.M.