MIXED PROBLEM FOR FREE RADIAL VIBRATIONS OF A CLOSED
SPHERICAL SANDWICH SHELL
V. Polyakov1, R. Chatys2
1
Latvian University, Institute of Polymer Mechanics
23 Aizkraukles, LV-1006, Riga, Latvia
e-mail: [email protected]
2
Technical University, Al. 1000-lecia Państwa Polskiego 7, Kielce, Poland
e-mail:[email protected]
ABSTRACT
Free vibrations across the thickness of a spherical shell are studied. The closed – form solution of
the one-dimensional wave problem was derived for a solid shell. A specific distinction of the radial
modes from those of radial vibrations of liquid was demonstrated. The solution was used in the
problem setting for a joint vibration of three spherical layers. The approximate version of vibrating
system based on the model approximation with the stiff and thin layers bordering the soft and thick
filler. The solution of dynamic problem for free vibrations of the soft filler was deduced by the
method of separation of variables. The elastic reactions of the thin face layers were taken as the
boundary conditions enabling three layers wave motion to be reduced to the midlayer motion under
mixed boundary conditions. The transcendental equation for the defined problem on free vibrations
was derived. The analysis and graphical description of the effect of thickness and elastic properties
on the eigenvalue spectrum that is distinctive of the mixed boundary problem was represented. The
curves were constructed for the main eigenfrequencies versus stiffness of the face layers and
geometric parameters that were characteristic of a sandwich shell.
1. Introduction
A study of interacting waves in the three-layered structures is a high-priority task to take into
account the radiated acoustic power from the structure excited by a acoustic wave. In an analytical
plan this problem entails inconvenience on the layered heterogeneous structure in the conditions of
wave dynamics. The developed enough approaches, based on variational principles, mainly apply
to the calculation of oscillation process for multi-layered plates and shells in the conditions of
Kirchhoff – Love’s classical kinematics for stiff layers with transversal shear in soft layers allowed for
[1,2]. The dispersion equations relating frequencies and wave numbers , mainly characterize wave
transmission in parallel with the plane of a plate or along the surface of a shell. The substantial role
of kinematic effect, related to the compression of sandwich along a normal due to transversal
compressibility of the core across the thickness has been analyzed in [3,4,5]. The analysis of
solutions was conducted for the problem of short-term loading on the edge of sandwich plate [3],
and the problem of acoustic shock on its surface [4]. High-frequency character of variation of the
asymmetric constituent of transversal displacements unlike a global deflection (a symmetric
constituent) varied in time was related to the local instability of thin face layers, i.e. with their
wrinkling from longitudinal force. A linear problem [5], involved the solution of wave equation within
the confines of the core, has been expressed in terms of displacements of the middle planes of two
facings. The latter were loaded with dynamic reactions from the side of liquid and the core.
Dispersion equation for estimation of wave numbers dependent on excitation frequency has been
derived using the coefficients of the set of bending equations.
Allowance for compressibility of a normal in all structural layers with optional elastic
parameters and geometric body under the joint dynamic behavior of layers requires the special
development of the solution to boundary-value problem for wave equations. For the structures of
sandwich type the incompressibility of a normal of the face layers with the finite values of Young’s
modules (by a factor of 103 greater then that of core) and relatively small thickness as compared
with core (by an order of magnitude and more less) can simplify the analysis of transverse vibration
and eigenfrequency spectrum of a sandwich particularly. High frequency of wave vibrations across
the thickness of the separate face layer is much superior to that of separate core. The eigenvalue
spectrum for a joint wave process in normal direction to all layers should be determined using the
solution of the contact dynamic interaction of three mediums different in the elastic properties. There
was pioneering work [6] thereupon, devoted to determination of oscillation process in
heterogeneous elements of structure, being in a dynamic contact. In order to motivate the simplified
assumption taken further in the present work at the construction of wave solution in the sandwich
structure the restored frequency equation of free vibrations of a bar of constant section, made from
two different in physical properties parts, is rewritten from [6]:
⎛l ω ⎞ ⎛l ω ⎞
tg ⎜⎜ 1 ⎟⎟ tg ⎜⎜ 2 ⎟⎟ =
⎝ c1 ⎠ ⎝ c 2 ⎠
where ci =
E1 ρ1
E2 ρ 2
(1)
Ei / ρ i , i = 1, 2 acoustic velocity on the area of a bar with the Young’s module Ei , a
specific density ρ i and length l i . The rigid fixing of a bar was accepted at the beginning of the first
part x = 0 whereas a free section was taken at the end of the second part x = l1 + l 2 . If module of
elasticity of the first part far exceeds that value for the other ( E1 >> E 2 ), then at E1 → ∞ from (1)
follows ω k = (c 2 / l 2 )(2k − 1)π / 2 , k = 1, 2,... Actually, the eigenfrequency of a bar at the specified
fixing is reduced to the frequency of free vibrations of the second part and no effect of the first part.
As the point of fixing of the bar changes, leaving free of loading the initial section of the first part
(with the inverse of number only in the right-hand part of the equation (1)), the following
transcendental equation after the aforecited limiting process results:
β 2 tgβ 2 =
l2 ρ 2
,
l1 ρ1
(β 2 =
ω l2
c2
)
(2)
Equation (2) is well known as determining eigenfrequencies of a bar with the adjoined mass at the
one end and fixed on the other. The notation of this equation for the second area with the very low
value of the modulus of elasticity as compared to the first part points out to the fact that
eigenfrequences of compound bar mainly depend on the acoustic velocity and the length of the
«weakest» link. Assuming the infinite value of acoustic velocity of the contiguous part with a free
end section (marked by an index 1) reduces to the account of the effect of this part of a bar on
frequency of joint vibrations as the adjoined mass. This trivial enough outcome on the static mass
inclusion (in sense of wave transmission in it with infinite velocity) is not so trivial allowing for threedimensional stress state in dynamic problems for layered plates and shells. The solution of
boundary-value problem for such structures becomes critically complicated with the introduction of
triaxial direction of inertial members and the account of curvature of the layered structure.
The most simplified variant solution of boundary-value dynamic problem and methodically
important for determination of eigenfrequency spectrum is given in the present work. A closed
spherical sandwich shell is examined for spherical waves propagation across the thickness of the
shell. A three-dimensional stress state in the layers is only governed by variation in radial
displacement component. The wave-induced radial vibrations of a sandwich are hypothetically
determined in terms of vibrations of the core with the use of deduced coefficients.
2. Radial vibrations of homogeneous sphere inclusive a central spherical cavity
Here we deduce the closed form of the solution for radial vibrations of the closed elasto-isotropic
sphere with a central spherical cavity. In this one-dimensional wave problem the volumetric stress
state and the eigenfrequency spectrum depend on the curvature of two bounding spherical
surfaces. Omitting the successive procedure to obtain the equation of motion in spherical
coordinates, we write out it in the final shape:
(1 − 2ν )(1 + ν ) ∂ 2 u
∂ 2 u 2 ⎛ ∂u u ⎞
⋅ 2 =0
+ ⎜ − ⎟− ρ
E (1 − ν )
∂t
∂r 2 r ⎝ ∂r r ⎠
(3)
where enter a radial displacement u ( r , t ) , the Young’s module E , a specific density ρ and the
Poisson’s coefficient ν . For the solution of wave equation (3) let us take advantage of variable
separation method admissive of the displacement expression u ( r , t ) = U ( r )T (t ) . This expression
has been substituted in Eq. (3) and having supposed that the expressions, depending on coordinate
r and t equal to constant − 1 / Λ2 , two equations have been obtained for the unknown functions:
T ′′(t ) +
E (1 − ν )
T (t ) = 0
Λ ρ (1 + ν )(1 − 2ν )
(4)
2
2
2⎞
⎛ 1
U ′′(r ) + U ′(r ) + ⎜ 2 − 2 ⎟U (r ) = 0
r
⎝Λ r ⎠
(5)
IIn equation (4) a differentiation with respect to t and in Eq. (5) with respect to r was marked by a
prime. The form of radial vibration that corresponds to a number Λ , has been derived from Eq. (4):
T (t ) = a sin(ω t ) + b cos(ω t )
(6)
The coefficients a, b should be derived from the initial conditions and a number Λ would be taken
from the countable eigenvalue spectrum, complying with the statement of boundary conditions. The
eigenfrequency of vibration in the form consistent with the number Λ k , equals
1
ωk =
Λk
⎡
⎤
E (1 − ν )
⎢ ρ (1 + ν )(1 − 2ν ) ⎥
⎣
⎦
1/ 2
,
k = 1, 2, ...
(7)
The standard solution procedure of equation (5) resulted in the general notation of solution in terms
of the cylindrical functions, including Bessel’s functions of order ± 3 / 2 . After application of recurrent
relations between Bessel’s functions their orders have been lowered by a unit, and having
considered the limits of summation of series for Bessel’s functions of order ± 1 / 2 the closed form
of a general solution of Eq. (5) shoul be found as:
⎛ sin( r / Λ ) cos (r / Λ ) ⎞
⎛ sin (r / Λ ) cos( r / Λ ) ⎞
⎟⎟
⎟⎟ + C 2 ⎜⎜
U (r ) = C1 ⎜⎜
+
−
2
Λr
r
r2
⎝ Λr
⎠
⎝
⎠
(8)
By a direct substitution of the solution (8) in Eq. (5) it is possible to make sure that the deduced form
of solution holds true.
The natural vibrations of a sphere with a central cavity were dependent on the boundary conditions
of fixing on the outside and the inside of the spherical body. The eigenvalues Λ were derived from
these conditions with the use of Eq. (8).
3. Determination of eigenfrequencies of a three-layer spherical shell
A layered spherical shell has considered further under fixedly connected heterogeneous layers (in
the sense of equality of the contact displacements and stresses). Free radial vibrations of three
spherical shells were characterized by a single eigenfrequency spectrum for all spherical areas. The
solutions of dynamic problem for three hollow spheres could not be «jointed» owing to the choice of
integration constants, as the frequency spectrum (7) has been obtained only for a separate sphere
and has been differed from two other ones. In order to overcome such a drawback it was possible to
take advantage of the approach, based on the consideration of substantial distinction between
elastic properties and thickness of face layers and a core. Hypothetically the vibrations of spherical
shell within the limits of the wall thickness of a sandwich structure were examined as a wave
process across the thickness of a core, “constrained” by face layers. The wave propagation across
the thickness of face layers has not been taken into account due to strong exceeding of values of
their modules of elasticity as compared with that of a core – by a factor of 102 - 103 . This admission
has been agreed on the estimates following from formulas (1), (2) for one – dimensional model of
vibrations. The next assumption was laid in the fact that the effect of face layers on the vibrations of
a core has been taken into account not as the adjoined mass, but as the elastic reactions of thin
spherical jackets on the upper and the lower surfaces of a core. The value of this reaction depended
on the radial displacements on contact surfaces and the elastic constants of face layers.
The coefficient of resistance on the contact boundary whereby the reaction affected the vibrations of
a core should be found as an absolute value of stress on the upper (or lower) surface of a thin
spherical zone to cause the unit radial displacement of a face layer on this boundary. Quasi-static
analysis of this stress should be in agreement with the assumption of an infinite velocity of radial
wave transmission in the face layer, under the stipulation that the time-dependent displacement has
been applied on its boundary surface.
R3
R2
R1
R4
Figure 1. A quarter of a normal section of three - layer sphere included a central spherical cavity.
The general solution of the equation (5) for 1 / Λ = 0 corresponded to quasi – static solution for the
displacement of a face spherical jacket. This simple solution u = B1r + B2 / r 2 has been set in the
expression of a radial stress σ r = (λ + 2 μ )(∂u / ∂r ) + 2λ (u / r ) and the latter has been rewritten as
σ r = 3kB1 − 4 μ B2 r 3 , where λ , μ and k have been constituted accordingly the Lame constants
and the module of volume elasticity.
One of two integration constants Bi , i = 1, 2 in the solution has been eliminated as the boundary
condition at the limiting radius R4 of the upper face layer and at R1 of the inner one has been
satisfied. see Fig.1. At the selected free boundary surface the boundary condition corresponded to
σ r = 0 , and in case of rigid fixing u = 0 . The remaining constant should be expressed in terms of
displacements and stresses at the contact surface with a core. Equating two expressions on the
contact surfaces the relationship between the displacement of a core and the response of a face
layer on each core surface should be found. The response p was directed oppositely to the
displacement and its value for the inner and the outer jackets shown in Fig.1 has been expressed in
terms of positive coefficients k12 and k 34 in that way:
p R = −k12 u ( R2 ),
p R = −k 34 u ( R3 )
2
(9)
3
where the coefficients obtained by above-mentioned procedure have appeared as
k12 =
1 − ( R1 R2 )3
12μ1k1
,
⋅
3
R2
4μ1 + 3k1 ( R1 R2 )
k34 =
( R4 R3 )3 − 1
12μ3k3
⋅
R3
4 μ3 + 3k3 ( R4 R3 )3
(10)
Lame constants and the modules of volume elasticity of sandwich layers could be expressed in
terms of Young modules and Poisson’s ratio of material, they have been marked respectively with
subscripts j = 1,3 for outer jackets and j = 2 for a core:
λj =
E jν j
(1 + ν j ) (1 − 2ν j )
, μj =
Ej
2(1 + ν j )
, kj =
Ej
3 (1 − 2ν j )
,
j = 1, 2, 3
(11)
The consequential solution of wave problem for a midlayer ( j = 2 ), as u 2 (r , t ) = U 2 (r ) T2 (t ) , has
involved the function of radial coordinate specified by the expression (8). The consequential
boundary conditions were expressed accordingly σ r R + u 2 R ⋅ k12 = 0 , σ r R + u 2 R ⋅ k 34 = 0 .
2
2
3
3
The substituends of stresses into the latter expressions have resulted in the boundary conditions as
connecting relations between the displacement and its normal derivative at core boundaries:
∂u 2
∂r
+ K 12 u 2
r = R2
r = R2
=0
∂u 2
∂r
+ K 34 u 2
r = R3
r = R3
=0
(12)
Coefficients K 12 , K 34 have been readily determined:
K12 =
12μ 3 k 3 [( R4 / R3 ) 3 − 1]
2ν 2
2ν 2
K
=
+
,
34
3
3
(λ 2 + 2μ 2 )[4μ1 + 3k1 ( R1 / R2 ) ] 1 − ν 2
(λ 2 + 2 μ 2 )[4μ 3 + 3k 3 ( R4 / R3 ) ] 1 − ν 2
12μ1k1[1 − ( R1 / R2 ) 3 ]
+
(13)
It should be noted that the mixed type of boundary conditions (12) must has also remained for the
problem of natural vibrations of homogeneous sphere with a spherical cavity, if its boundary
surfaces had free, that was meant for them σ r
R
= 0 . In this case there were only the second terms
in the expressions of coefficients (13) that must have remained.
Omitting the successive procedure to obtain the eigenvalues of the considered boundary - value
problem for free radial vibrations of tree-layered spherical shell let us denote the main aspects of
calculations. The displacement as Eq. (8) and its derivative, deduced at r = R2 and r = R3 , has
been substituted in equations (12). The substitution has brought the boundary conditions to the set
of four homogeneous equations. The elementary calculations, related mainly to transformation of
trigonometric functions included in the expression for a main determinant, have result in the
following transcendental equation deciding the eigenvalues:
tgξ =
a3ξ 3 + ξ
b4ξ 4 + b2ξ 2 + 1
,
(14)
where ξ = ( R3 − R2 ) / Λ , and the coefficients of partial fraction expansion in the right-hand part
depended on the radiuses of a midlayer and the given parameters (13) determining elastic
resistance of face layers at the conjugation boundary with a core:
a3 =
R2 R3
( R3 − R2 ) 3
⎛ R3
R2 R3
R2 ⎞
⎜⎜
⎟⎟ , b2 =
−
( R3 − R2 ) 2
⎝ 2 − K 34 2 − K12 ⎠
b4 =
⎡
⎤
R3
R2
−
⎢1 −
⎥
⎣ R3 (2 − K12 ) R2 (2 − K 34 ) ⎦
R22 R32
( R3 − R2 ) 4 (2 − K12 )(2 − K 34 )
(15)
The denumerable set of roots ξ k , k = 1, 2,... in a positive semi-axis ξ is determined by the
abscissas of points where branches of denumerable set of tangent curves, located along this axis,
and the curve, determined by the expression in right-hand part of (15) have crossed each other. The
eigenvalues of boundary-value problem were further calculated taking into account the thickness of
a midlayer:
Λ k = ( R3 − R2 ) / ξ k ,
k = 1, 2,...
(16)
The eigenfrequences of vibrations of the spherical layer of a core, “edged” with thin face jackets,
were determined in terms of parameters of the elasticity of midlayer and the eigenvalues Λ k .
Following a general notation for Eq. (7) the eigenfrequences can be found
ξk
⎡
⎤
E 2 (1 − ν 2 )
ωk =
⎢
⎥
2π ( R3 − R2 ) ⎣ ρ 2 (1 + ν 2 )(1 − 2ν 2 ) ⎦
1/ 2
,
k = 1, 2,...
(17)
4. Variants of calculations for eigenfrequences attributed to a choice of structure
From Eq. (14) follows that at large values ξ the ratio of polynomials tends to zero. The roots of the
equation at the indicated asymptotic behavior of its right-hand part equal ξ k = kπ , when k → ∞ .
Of most interest were the initial root values, in particular, the least values corresponding to master
frequency of free vibrations.
Variation of functions f i (ξ ) which represented the left (i = 1) and the right part (i = 2,..,5) of
equation (14) at the different parameters of structure has shown in Fig. 2, a, b accordingly on the
initial interval of variation of parameter ξ and around a number π . Structure parameters for the
wall of a spherical shell of sandwich type ( i = 5 ) were resulted in Table 1.
TABLE 1. Parameters of the sandwich three-layer sphere.
R1 = 100 mm
Layer
R4 − R3
R3 − R2
30
R2 − R1
5
ν
E , GPa
5
ρ , kg / m 3
10
0.30
_
0.04
0.40
119
10
0.30
_
By transformation of structural model, related to change in its geometrical parameters, the variations
of corresponding functions has been got for a homogeneous shell with a free boundary
surfaces (i = 2,3) . At degeneration of the elastic constants of face layers in case of infinite rigidity of
spherical jackets, the model of sandwich is reduced to the homogeneous shell of a core with the
rigid fixing of its boundary surfaces (i = 4) .
a
1.5
b
fi
0.8
1
fi
1
0.6
2
1
0.4
0.2
3
3
0.5
4
0
2
ξ
4
0
ξ
0.2
5
5
2.83
3
3.17
3.34
3.51
3.68
3.85
Figure 2. Variation of functions f i (ξ ), i = 1,...,5 in the eigenvalue equation (14) over a - the initial range of
ξ = ( R3 − R2 ) / Λ and b - within the interval 0.9π ≤ ξ ≤ 1.2π .
The equation for determination of eigenvalues of radial vibrations of a homogeneous spherical shell
with free surfaces has been automatically deduced from Eq. (14):
R2 R3ν ∗
⋅ξ 3
2
( R3 − R2 )
,
tgξ =
2
2
( R2 + R3 )ν ∗ − R2 R3 2
R22 R32ν ∗2
4
⋅ξ +
⋅ξ
1−
( R3 − R2 ) 2
( R3 − R2 ) 4
ξ+
where ν ∗ =
1 −ν 2
2(1 − 2ν 2 )
(18)
and similarly in the case of clamped spherical surfaces of a homogeneous shell it has been found:
tgξ =
ξ
R2 R3
⋅ξ 2
1+
2
( R3 − R2 )
(19)
The calculated values of eigenfrequencies and eigenvalues for tree-layer spherical shell with a
central cavity and homogeneous ball were represented in Figure 3 a, b.
a
14
ω1
12
b
⋅ 10 − 3 , Hz
5000
___
4000
....... ξ ⋅ 10
1
10
ω 1 , Hz
3
1
3000
2
8
2000
3
6
ξ
4
π
2
1000
32
34
Rout , mm
5
1
h , mm
30
4
36
38
0
150
40
Figure 3. Variation of the reference frequency
ω1 , Hz
200
and matching eigenvalue
250
ξ1
300
for radial vibrations of a
three layer spherical shell versus h = R3 − R2 (a) and for a solid sphere versus outer radius Rout (b).
The eigenfrequency curves 1,2,3 in Figure 3 b were obtained versus outer radius Rout for solid balls
whereas the curves 4,5 corresponded to a shell with a spherical cavity of radius Rinn = 50 mm . The
rigidly fixed bounding surfaces corresponded to first curve, the free surfaces – curves 2,4 at
ν = 0.4 , and 3 at ν = 0.3 . It should be noted that for the problem of radial vibrations of elastic
monolithic sphere with a fixed surface the equation of eigenvalues was derivable from (19) as
tgξ = ξ with the least nonzero root value that equals ξ1 = 4.493 . The same equation holds true for
evaluation of eigenvalues of a boundary-value problem for radial vibrations of gas in a spherical
vessel [7].
5. Conclusions
An analytical model for free radial vibration of a closed three-layer spherical shell has been derived.
The elastic compliance of a middle layer and its greater thickness as compared with outer layers
have been used as the main feature of model making. The main working-out and the main results
can be outlined as:
•
The elastic reactions of the thin face layers were taken as the boundary conditions enabling
three layers wave motion to be reduced to the vibration of the midlayer under mixed boundary
conditions. The transcendental equation of eigenvalues for the defined problem on free
vibrations has been derived.
•
The variation of eigenvalues as well as corresponding eigenfrequencies have been examined
versus curvatures and thickness of a spherical shell. In particular it has been found that the roots
of the eigenvalue equation for a shell of sandwich structure fell into the small vicinity of a number
π , unlike a homogeneous shell with the free boundary surfaces.
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