On equilibrium equations of cylindrical shell with attached rigid body
The mechanical system consisting of a circular cylindrical shell and a rigid body attached to one of the shell ends is considered. In linear statements, the boundary-value problem on a stressedlydeformed state of this system under concentraited and distributed loads is formulated. The equations ob...
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| Cite this: | On equilibrium equations of cylindrical shell with attached rigid body / Y.V. Trotsenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 422-431. — Бібліогр.: 4 назв. — англ. |
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| author_facet | Trotsenko, Y.V. |
| citation_txt | On equilibrium equations of cylindrical shell with attached rigid body / Y.V. Trotsenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 422-431. — Бібліогр.: 4 назв. — англ. |
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| description | The mechanical system consisting of a circular cylindrical shell and a rigid body attached to one
of the shell ends is considered. In linear statements, the boundary-value problem on a stressedlydeformed state of this system under concentraited and distributed loads is formulated. The
equations obtained can also be used for a study of free oscillations of the considered construction if one replaces the applied loads with forces of inertia and their moments.
|
| first_indexed | 2025-12-07T16:00:40Z |
| format | Article |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 3, 2001
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL
WITH ATTACHED RIGID BODY
Yu. V. Trotsenko
Interbranch Research Institute of Mechanics Problems “Ritm”,
pr. Povitryanoflots’ky, 37, Kyiv, 03056, Ukraine
The mechanical system consisting of a circular cylindrical shell and a rigid body attached to one
of the shell ends is considered. In linear statements, the boundary-value problem on a stressedly-
deformed state of this system under concentraited and distributed loads is formulated. The
equations obtained can also be used for a study of free oscillations of the considered constructi-
on if one replaces the applied loads with forces of inertia and their moments.
AMS Subject Classification: 70G75, 74 G10
1. The Equations of Equilibrium and the Boundary Conditions for the System “Body-
Shell”
Today, the necessity of calculation of the stressedly-deformed state and dynamical characteri-
stics of the mechanical system consisting of a circular cylindrical shell and a rigid body attached
to one of the shell ends arises in different fields of science and engineering. Below a linear
mathematical model of equilibrium state of the considered mechanical system under loads
of general form is consructed on the base of the shell theory proposed by V. Z. Vlasov. The
equations of the disturbed state of a prestressed, flexible, rotational shell with a rigid concentric
inclusion in the form of elastic disk are obtained in paper [1].
Let us consider a mechanical system consisting of a thin, circular, cylindrical shell and
a perfactly rigid body which is rigidly attached to one of the shell ends. Suppose that the
other shell end is fixed in a certain way. Let the body have two symmety planes whose line of
intersection, Oz coincides with the shell longitudinal axis. The coordinate plane Oxz is combi-
ned with one of the symmetry planes of the rigid body, and the originO of the coordinate system
Oxyz is placed in the plane of the end section of the shell which is free of the rigid body.
To describe the body displaycements let us introduce an orthogonal coordinate system
Cxcyczc with its origin C placed at the center of inertia of the body. Its axes, Cxc and Cyc, are
parallel to the axes Ox and Oy respectivly. The unit vectors of the coordinate system Cxcyczc
are denoted by~ic,~jc,~kc. A median surface of the cylindrical shell is referred to the orthogonal
curvilinear coordinate system of z and ϕ, where ϕ is the polar angle counted from Ox axis.
With this coordinate system we connect a local orthogonal basis ~e1, ~e2, ~e3 the unit vectors ~e1,~e2
of which are tangent to the principal curvature lines of the shell’s median surface and are di-
rected in direction of increase of the coordinates z and ϕ. The vector ~e3 = [~e1 × ~e2].
Suppose that the considered construction is under a small load of the most general form,
namely the rigid body is under the force ∆~F = ∆F1
~ic + ∆F2
~jc + ∆F3
~kc concentraited in the
point C and under the moment relative to the point C, ∆ ~M = ∆M1
~ic + ∆M2
~jc + ∆M3
~kc.
In turn, the shell is under a distributed load ∆ ~Q = ∆Q1~e1 + ∆Q2~e2 + ∆Q3~e3. As a result the
system will come to a disturbed equilibrium state and be subjected to strains and displacements.
We shall characterize this equilibrium state by the displacement vector of points of the shell’s
422 c© Yu. V. Trotsenko, 2001
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL WITH ATTACHED RIGID BODY 423
median surface, ~u = u~e1 + v~e2 + w~e3, by the displacement vector of the center of mass of
the body, ~u0 = u01
~ic + u02
~jc + u03
~kc, and by the vector of turning angle around this center,
~θ0 = θ01
~ic + θ02
~jc + θ03
~kc. In addition, we suppose that the displacements of the rigid body and
the shell are so small that one can neglect the terms of the second and higher order of smallness
in comparison with the terms of the first order.
To describe the stressedly-deformed state of the cylindrical shell, we shall use the shell
theory based on the Kirchhoff – Love hypotheses. The moment shell theory based on this hypo-
theses is successfully applied for solving statics and dynamics problems. But one must be careful
in applying this theory, since the corresponding boundary-value problem may be not self-adjoint
and thus an input problem may not be formulated in the form of the corresponding vari-
ational principle. In this case, when the problem of free oscillations is solved, one cannot
guarantee the reality of the natural frequencies. In addition, the difficulties arise when one
formulates the conditions of orthogonality of the natural modes which play an essential role
in calculation of shell constructions and in investigation of a response of this constructions
to arbitrary perturbations. These difficulties may be overcomed within the framework of the
Kirchhoff – Love hypotheses as well by means of writing equations that would lead to a self-
adjoint boundary problem. For this purpose it is necessary to choose a certain variant of the
elastic relations that do not contradict the sixth equation of equilibrium [2]. But such an
approach leads to a complication of equations of the shell theory that hampers the constructi-
on of their solutions. In this sense, a variant of the engineering shell theory worked out by V.
Z. Vlasov [3] is more preferable. It is the simplest shell theory which, along with a satisfactory
accuracy, leads to the self-adjoint boundary-value problems that permits to obtain equations of
this theory from variational principle. The last circumstance opens perspectives in applications
of the energy method for solving the formulated boundary-value problems.
Thus, according to the engineering thin shell theory the tangential components of the bend
deformations is neglected and the equilibrium equations of a cylindrical shell element are
represented as follows [2]:
∂T1
∂z
+
1
R
∂S
∂ϕ
+ ∆Q1 = 0,
∂S
∂z
+
1
R
∂T2
∂ϕ
+ ∆Q2 = 0, (1)
D∆∆W +
1
R
T2 − ∆Q3 = 0,
where
D =
Eh3
12 (1 − ν2)
, ∆ = R2 ∂
2
∂z2
+
∂2
∂ϕ2
;
E, ν, h, and R is the elastic module, the Poisson’s ratio, the thickness and the radius of the shell;
T1 , T2, and S is the meridional force, the ring force, and the shering force, which is refered to
the unit of length of the normal section of the shell’s median surface.
424 Yu.V. TROTSENKO
The forces and the moments in the normal sections of shell are associated with the compo-
nents of a median surface deformation and with the parameters of change of the median surface
curvature by
T1 =
Eh
1 − ν2
(ε1 + νε2) , T2 =
Eh
1 − ν2
(ε2 + νε1) , S =
Eh
2 (1 + ν)
ω,
(2)
M1 = D (χ1 + νχ2) , M2 = D (χ2 + νχ1) , M12 = D (1 − ν)χ12,
whereM1, M2, M12 is the linear bending moment in the meridional plane, the linear peripheral
moment, and the linear torque, respectively.
The linear lateral force must by calculated with by the formula
Q1 = D
[
(1 − ν)
R
∂χ12
∂ϕ
+
∂χ1
∂z
+ ν
∂χ2
∂z
]
. (3)
In turn, six components of the median surface deformation of the shell are expressed in
terms of its displacements in the following way:
ε1 =
∂u
∂z
, ε2 =
1
R
(
∂v
∂ϕ
+ w
)
, ω =
1
R
∂u
∂ϕ
+
∂v
∂z
,
(4)
χ1 = −∂
2w
∂z2
, χ2 = − 1
R2
∂2w
∂ϕ2
, χ12 = − 1
R
∂2w
∂z∂ϕ
.
If in equations (1) one expresses the forces according to elastic equations (2) and replaces
deformations for displacements (4), the equilibrium equations of the circular cylindrical shell
in displacements will be obtained. It is convenient to reduce this system of equations to the
following matrix form:
L~u = ~g. (5)
Here
L =
∥∥∥∥∥∥
L11 L12 L13
L21 L22 L23
L31 L32 L33
∥∥∥∥∥∥ , ~u =
∥∥∥∥∥∥
u
v
w
∥∥∥∥∥∥ , ~g =
1 − ν2
Eh
∥∥∥∥∥∥
−∆Q1
−∆Q2
∆Q3
∥∥∥∥∥∥ ,
L11 =
∂2
∂z2
+
ν1
R2
∂2
∂ϕ2
, L12 = L21 =
ν2
R
∂2
∂z∂ϕ
, L13 = L31 =
ν
R
∂
∂z
,
L22 =
1
R2
∂2
∂ϕ2
+ ν1
∂2
∂z2
, L23 = L32 =
1
R2
∂
∂ϕ
, L33 =
1
R2
(
c2∆∆ + 1
)
,
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL WITH ATTACHED RIGID BODY 425
c2 =
h2
12R2
, ν1 =
1 − ν
2
, ν2 =
1 + ν
2
.
Let us turn to deriving the equilibrium equations of the rigid body. To this end, let us
calculate the forces and the moments, relative to the point C, which act on the body. The
elastic forces and moments (referenced to the unit of length of the section of the shell’s median
surface) will act on the edge of the cylindrical shell; they are of the following form [2]:
~T = − (T1~e1 + S~e2 +Q1~e3) , ~M = M12~e1 −M1~e2.
However, by analogy with the bending plate theory and starting from the Kirchhoff kinematic
hypothesis, one can establish that the torque M12 on the shell edge is statically equivalent to
the lateral force distributed along the contour and its intensity is described by the expression
R−1∂M12/∂ϕ [4]. With regard to this circumstance, the linear forces and the moments acting
on the body and coming from the shell will be equal to
~T = − (T1~e1 + S~e2 +Q∗
1~e3) , ~M = −M1~e2. (6)
Here Q∗
1 is a generalized lateral force on the shell contour calculated, with regard to relations
(3) and (4), according to the formula
Q∗
1 = Q1 +
1
R
∂M12
∂ϕ
= −c2 Eh
1 − ν2
[
R2∂
3w
∂z3
+ (2 − ν)
∂3w
∂z∂ϕ2
]
. (7)
Taking into account the relation between the Darboux unit vectors and the unit vectors of
the coordinate system Cxcyczc, which are of the form
~e1 = ~kc,
~e2 = − sinϕ~ic − cosϕ~jc, (8)
~e3 = cosϕ~ic − sinϕ~jc,
we shall represent vectors (6) as expansions relatively to the unit vectors~ic,~jc,~kc. In this case
we shall have
~T = (S sinϕ−Q∗
1 cosϕ)~ic + (Q∗
1 sinϕ+ S cosϕ)~jc − T1
~kc,
~M = M1 sinϕ~ic +M1 cosϕ~jc.
Denote futher the distance along the axis Oz from the point C to the shell end section to
which the rigid body is attached by lc. Then for the resulting moment ~My
c , which is relative to
426 Yu.V. TROTSENKO
point C, of the elastic forces acting on the rigid body, we shall obtain the following expression
My
c =
∮
L
[
~r0 × ~T
]
ds = ~ic
∮
L
[T1R sinϕ+ lc (Q∗
1 sinϕ+ S cosϕ)] ds
+ ~jc
∮
L
[T1R cosϕ+ lc (Q∗
1 cosϕ− S sinϕ)] ds+ ~kc
∮
L
RSds, (9)
where ~r0 = (R cosϕ)~ic − (R sinϕ)~jc − lc~kc is the radius vector of points of the shell’s edge
contour in the coordinate system Cxcyczc; L is the contour formed by the cross-section of the
median shell surface at z = l; s is the length of the contour arc; l is the length of the cylindrical
shell. After calculation of the resulting vector of all forces acting on the rigid body we arrive at
the following three scalar relations:∮
L
(Q∗
1 cosϕ− S sinϕ) ds = ∆F1,
∮
L
(Q∗
1 sinϕ+ S cosϕ) ds = −∆F2, (10)
∮
L
T1ds = ∆F3.
Similarly, with regard to expression (9) and using the condition that the resulting moment
relative to the center of mass of the rigid body equals zero, we shall obtain another equations,
namely, ∮
L
[T1R sinϕ+M1 sinϕ+ lc (Q∗
1 sinϕ+ S cosϕ)] ds = −∆M1,
∮
L
[M1 cosϕ+ T1R cosϕ+ lc (Q∗
1 cosϕ− S sinϕ)] ds = −∆M2, (11)
∮
L
SRds = −∆M3.
The boundary conditions on the shell contour at z = 0 should be added to equations (5),
(10), (11), as well as the relations connecting the displacements and the angles of rotation of
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL WITH ATTACHED RIGID BODY 427
the shell with the corresponding generalized coordinates of the rigid body in the place of the
shell and body binding.
The equality of the displaycements of the shell and the rigid body on the contour L leads to
the relation
~u = ~u0 +
[
~θ0 × ~r0
]
. (12)
Let us express the right-hand side of expression (12) in the form an expansion with respect to
the unit vectors of the Darboux trihedron, taking into account their connection with the unit
vectors of the coordinate system Cxcyczc, in the form of
~ic = − sinϕ~e2 + cosϕ~e3,
~jc = − cosϕ~e2 − sinϕ~e3, (13)
~kc = ~e1.
After equating the vectors components in expression (12) with regard to (13) we shall obtain
u = u03 − θ01R sinϕ− θ02R cosϕ,
v = (θ02lc − u01) sinϕ− (θ01lc + u02) cosϕ− θ03R, (14)
w = − (θ01lc + u02) sinϕ+ (u01 − θ02lc) cosϕ.
Boundary conditions on the shell end are imposed on the displacements u, v, w and on the
angle of rotation, θ1 = −∂w/∂z, of the vector ~e1 around the vector ~e2 as a result of deformation
of the shell’s median surface. In order to determine the corresponding angle of rotation of the
rigid body, we shall calculate the vector product
[
~kc × ~k∗
]
, where ~k∗ is a unit vector of the body
coordinate system Cxcyczc directed along the axis C∗z∗. Within the accuracy of linear terms,
this unit vector is equal to
~k∗ = θ02
~ic − ~θ01
~jc + ~kc. (15)
With regard for relations (15) and (13) we shall have[
~kc × ~k∗
]
= θ01
~ic + θ02
~jc = (−θ02 cosϕ− θ01 sinϕ)~e2 + (−θ02 sinϕ+ θ01 cosϕ)~e3.
Equating the angle of rotation of the vector ~kc around the direction ~e2 to the corresponding
angle of rotation of the shell edge, we shall obtain
∂w
∂z
∣∣∣∣
z=l
= θ01 sinϕ+ θ02 cosϕ. (16)
Thus determination of the disturbed state of the considered system is reduced to solving
shell equations (5) together with the body equilibrium equations (10), (11) subject to conditions
(14), (16) on the contourL. Conditions of binding of the shell end, which is free of the rigid body,
should be added to this relations.
428 Yu.V. TROTSENKO
2. Derivation of the Equilibrium Equations from the Variational Virtual Displaycements
Principle
In previous section an integro-differential statement of the problem on determining the
equilibrium state of a cylindrical shell with an attached rigid body is given under the most
general small load. In the present section, the equilibrium equations of this mechanical system
will be derived with the use of the variational principles of mechanics. Such an approach will
serve as an additional criterion of reliability of the constructed mathematical model and besides
will permit to formulate an equivalent variational statement of the considered problem which
can later on be used for constructing an approximate solution of this problem.
To obtain the equilibrium equations of the considered system and natural boundary condi-
tions, we use the virtual displycements principle according to which
δΠ = δA, (17)
where δΠ is a variation of the potential energy of the system; δA is a variation of the work of
external forces.
The work of external forces acting on the body and the shell is equal to
A =
∫∫
Σ
∆ ~Q · ~udΣ + ∆~F · ~u0 + ∆ ~M · ~θ0, (18)
where Σ is the median surface of shell.
The potential strain energy of a thin cylindrical shell may be represented in the form [2]
Π =
Eh
2 (1 − ν2)
∫∫
Σ
[
(ε1 + ε2)2 − 2 (1 − ν)
(
ε1ε2 −
ω2
4
)]
dΣ
+
D
2
∫∫
Σ
[
(χ1 + χ2)2 − 2 (1 − ν)
(
χ1χ2 − χ2
12
)]
dΣ. (19)
The first term in formula (19) is the potential energy of elongation and shear and the second
is the potential energy of bending and torsion. Substituting the expressions of the components
of the strain of the shell median surface (4) into (19) we obtain the following expression of the
potential energy in displaycements
Π =
Eh
2 (1−ν2)
∫∫
Σ
[(
∂u
∂z
)2
+
1
R2
(
∂v
∂ϕ
+w
)2
+
2ν
R
∂u
∂z
(
∂v
∂ϕ
+w
)
+
1−ν
2
(
1
R
∂u
∂ϕ
+
∂v
∂z
)2
]
dΣ
+
D
2
∫∫
Σ
[(
∂2w
∂z2
)2
+
(
1
R2
∂2w
∂ϕ2
)2
+
2ν
R2
∂2w
∂z2
∂2w
∂ϕ2
+ 2 (1 − ν)
(
1
R
∂2w
∂z∂ϕ
)2
]
dΣ. (20)
Denote the displaycement variations of points of the shell’s median surface by δu, δv, δw.
Then the variation of potential energy of the shell elastic strain takes the following form:
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL WITH ATTACHED RIGID BODY 429
δΠ =
Eh
1 − ν2
∫∫
Σ
{[
∂u
∂z
+
ν
R
(
∂v
∂ϕ
+ w
)]
∂δu
∂z
+
[
ν
R
∂u
∂z
+
1
R2
(
∂v
∂ϕ
+ w
)]
∂δv
∂z
+
[
ν
R
∂u
∂z
+
1
R2
(
∂v
∂ϕ
+ w
)]
δw + ν1
(
1
R2
∂u
∂ϕ
+
1
R
∂v
∂z
)
∂δu
∂ϕ
+ ν1
(
1
R
∂u
∂ϕ
+
∂v
∂z
)
∂δv
∂z
}
dΣ +D
∫∫
Σ
[(
∂2w
∂z2
+
ν
R2
∂2w
∂ϕ2
)
∂2δw
∂z2
+
(
1
R4
∂2w
∂ϕ2
+
ν
R2
∂2w
∂z2
)
∂2δw
∂ϕ2
+
2 (1 − ν)
R2
∂2w
∂z∂ϕ
∂2δw
∂z∂ϕ
]
dΣ. (21)
In what follows we shall suppose that the shell end, which is free of the rigid body, is rigidly
fixed. For the functions f (z, ϕ) and g (z, ϕ) which are 2π-periodic with respect to ϕ one can
establish, on condition that g (0, ϕ) = 0, the following formulae of integration by parts for the
surface integrals
∫∫
Σ
f
∂g
∂z
dΣ = −
∫∫
Σ
g
∂f
∂z
dΣ +
∮
L
fgds,
(22)∫∫
Σ
f
∂g
∂ϕ
dΣ = −
∫∫
Σ
g
∂f
∂ϕ
dΣ.
Applying formula (22) to integrals in (21), we get rid of derivatives of the variations δu, δv, δw
in them. Taking into account that the displacement variations satisfy the principal boundary
conditions on the shell contour at z = 0, expression (21) may be reduced to the form
δΠ = − Eh
1−ν2
∫∫
Σ
[(
∂2u
∂z2
+
ν2
R
∂2v
∂z∂ϕ
+
ν
R
∂w
∂z
+
ν1
R2
∂2u
∂ϕ2
)
δu−
(
1
R2
∂v
∂ϕ
+
1
R2
w+
ν
R
∂u
∂z
)
δw
+
(
1
R2
∂2v
∂ϕ2
+
1
R2
∂w
∂ϕ
+
ν2
R
∂2u
∂z∂ϕ
+ ν1
∂2v
∂z2
)
δv
− c2
R2
(
R4∂
4w
∂z4
+
∂4w
∂ϕ4
+ 2R2 ∂4w
∂z2∂ϕ2
)
δw
]
dΣ
430 Yu.V. TROTSENKO
+
Eh
1 − ν2
∮
L
{[
∂u
∂z
+
ν
R
(
∂v
∂ϕ
+ w
)]
δu+ ν1
(
1
R
∂u
∂ϕ
+
∂v
∂z
)
δv
}
ds
+
D
R2
∮
L
{[
−R2∂
3w
∂z3
− (2 − ν)
∂3w
∂z∂ϕ2
]
δw +
(
R2∂
2w
∂z2
+ ν
∂2w
∂ϕ2
)
∂δw
∂z
}
ds. (23)
On the contour L the variations δu, δv, δw, and ∂δw/∂z are not independent, since the
shell displacements are connected with parameters of the rigid body motion by the conjucti-
on conditions. Taking into account the expressions of the shell displacement variations on the
contour L, we represent the variational equation (17) in the form
− Eh
1 − ν2
{∫∫
Σ
[(
∂2u
∂z2
+
ν2
R
∂2v
∂z∂ϕ
+
ν
R
∂w
∂z
+
ν1
R2
∂2u
∂ϕ2
+
1 − ν2
Eh
∆Q1
)
δu
+
(
1
R2
∂2v
∂ϕ2
+
1
R2
∂w
∂ϕ
+
ν2
R
∂2u
∂z∂ϕ
+ ν1
∂2v
∂z2
+
1 − ν2
Eh
∆Q2
)
δv
−
(
1
R2
∂v
∂ϕ
+
1
R2
w +
ν
R
∂u
∂z
+
c2
R2
∆∆w − 1 − ν2
Eh
∆Q3
)
δw
]
dΣ
}
+
∮
L
(Q∗
1 cosϕ− S sinϕ) ds− ∆F1
δu01
+
∮
L
(Q∗
1 sinϕ+ S cosϕ) ds+ ∆F2
δu02 +
∮
L
T1ds− ∆F3
δu03
+
∮
L
(RT1 sinϕ+M1 sinϕ+ lcS cosϕ+ lcQ
∗
1 sinϕ) ds+ ∆M1
δθ01
+
∮
L
(M1 cosϕ+RT1 cosϕ−lcS sinϕ+lcQ
∗
1 cosϕ) ds+∆M2
δθ02
+
∮
L
RSds+∆M3
δθ03=0. (24)
Equating the coefficients of δu, δv, δw in the surface integrals to zero and using the indepen-
dence of the shell displacement variations and the variations of parameters of the rigid body
motion, we shall obtain the equilibrium equations of a cylindrical shell (5). In turn, setting the
ON EQUILIBRIUM EQUATIONS OF CYLINDRICAL SHELL WITH ATTACHED RIGID BODY 431
coefficients of the variations of the rigid body parameters equal to zero gives us the equilibrium
equations (10), (11). As a result the boundary-value problem for the considered system which
was formulated in the previous section turned out to be equivalent to the variational equa-
tion (17).
It should be noted separatly that the equilibrium equations (10) and (11) are the natural
boundary conditions for the functional I = Π − A on the class of functions which satisfy the
conjugation conditions (14), (16) and the fixing conditions of the shell end which is free of the
rigid body. It means that in minimization of the functional I on the mentioned function class
the necessity of a priori realization of rather complicated boundary conditions (10) and (11) is
eliminated. This gives a certain advantage to the energy method of construction of an approxi-
mate solution for the considering problem in comparison with other methods of mathematical
physics.
The equations obtained can be used to determine the small strains of the cylindrical shell
and displacements of the rigid body under a small load. This equations can also be used to study
free oscillations of the considered system if according to the d’Alambert principle, one carries
out the change of the load components in the equilibrium equations for the corresponding
forces of inertia and their moments.
REFERENCES
1. Kladinoga V.S. and Trotsenko V.A. “On equations of disturbed state of a prestrained flexible shell with
attached rigid body,” Dokl. Akad. Nauk Ukraine, No. 5, 61 – 65 (1992).
2. Novozhilov V.V. Thin Shell Theory [in Russian], Sudpromgiz, Leningrad (1962).
3. Vlasov V.Z. Selected Works, Vol. 1 [in Russian], Akad. Nauk SSSR, Moscow (1962).
4. Biderman V.L. Mechanics of Thin-Shell Constructions [in Russian], Mashinostroenie, Moscow (1977).
Received 20.05.2000
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| id | nasplib_isofts_kiev_ua-123456789-174701 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-3076 |
| language | English |
| last_indexed | 2025-12-07T16:00:40Z |
| publishDate | 2001 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Trotsenko, Y.V. 2021-01-27T11:39:04Z 2021-01-27T11:39:04Z 2001 On equilibrium equations of cylindrical shell with attached rigid body / Y.V. Trotsenko // Нелінійні коливання. — 2001. — Т. 4, № 3. — С. 422-431. — Бібліогр.: 4 назв. — англ. 1562-3076 AMS Subject Classification: 70G75, 74 G10 https://nasplib.isofts.kiev.ua/handle/123456789/174701 The mechanical system consisting of a circular cylindrical shell and a rigid body attached to one of the shell ends is considered. In linear statements, the boundary-value problem on a stressedlydeformed state of this system under concentraited and distributed loads is formulated. The equations obtained can also be used for a study of free oscillations of the considered construction if one replaces the applied loads with forces of inertia and their moments. en Інститут математики НАН України Нелінійні коливання On equilibrium equations of cylindrical shell with attached rigid body Про рівняння рівноваги циліндрічної оболонки з приєднаним твердим тілом Об уравнениях равновесия цилиндрической оболочки с присоединенным твердым телом Article published earlier |
| spellingShingle | On equilibrium equations of cylindrical shell with attached rigid body Trotsenko, Y.V. |
| title | On equilibrium equations of cylindrical shell with attached rigid body |
| title_alt | Про рівняння рівноваги циліндрічної оболонки з приєднаним твердим тілом Об уравнениях равновесия цилиндрической оболочки с присоединенным твердым телом |
| title_full | On equilibrium equations of cylindrical shell with attached rigid body |
| title_fullStr | On equilibrium equations of cylindrical shell with attached rigid body |
| title_full_unstemmed | On equilibrium equations of cylindrical shell with attached rigid body |
| title_short | On equilibrium equations of cylindrical shell with attached rigid body |
| title_sort | on equilibrium equations of cylindrical shell with attached rigid body |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/174701 |
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