Download Longitudinal Waves in a Rotating Solid Cylinder Immersed in an

Frontiers of Engineering Mechanics Research
May 2013, Vol. 2 Iss. 2, PP. 42-50
Longitudinal Waves in a Rotating Solid Cylinder
Immersed in an Inviscid Fluid Using Chebyshev
Polynomial
R. Selvamani*1, P.Ponnusamy2
1
Department of Mathematics, Karunya University, Coimbatore, TamilNadu, India
Department of Mathematics, Govt Arts College, Coimbatore, TamilNadu, India
*1
[email protected]
2
Abstract-The longitudinal waves in a homogeneous transversely isotropic rotating solid cylinder immersed in an in viscid fluid is
studied using the linearized, three-dimensional theory of elasticity. The equations of motion of solid and fluid are respectively
formulated using the constitutive equations of a transversely isotropic cylinder and the constitutive equations of an in viscid fluid. The
solution of the frequency equations are obtained by Chebyshev polynomials series using the geometric boundary conditions. The
computed non-dimensional frequencies are presented in the form of dispersion curves for the material Zinc.
Keywords- Solid Cylinder Loaded with Fluid; Free Vibration of Transversely Isotropic Cylinder; Wave Propagation in a Rotating
Cylinders; Chebyshev Polynomials
I.
INTRODUCTION
Solid cylinder immersed in fluid is the practical element of many types of engineering structure such as marine craft and
airplanes in which the machine induced vibrations often occurs. It is therefore very important that the static and dynamic
behavior of plates when subjected to different loading conditions be clearly understood so that they may be safely used in these
industrial applications. It is well known that the natural frequencies of structures in contact with fluid are different from those
in vacuo. The longitudinal modes often used to evaluate the material properties of thin metal wires, reinforcement filament in
ultrasonic transducers and resonators. Applying Chebyshev polynomial series as the admissible function for each
displacement component has distinct advantages like rapid convergence and better numerical stability in computation than
other algebraic polynomial series.
The most general form of harmonic waves in a hollow cylinder of circular cross section of infinite length has been analyzed
by Gazis [1]. Mirsky [2] investigated the wave propagation in transversely isotropic circular cylinders of infinite length and
presented the frequency equation in Part I and numerical results in Part II. A method, for solving wave propagation in arbitrary
cross-sectionl cylinders and plates and to find out the phase velocities in different modes of vibrations namely longitudinal,
torsional and flexural, by constructing frequency equations was devised by Nagaya [3-5]. He formulated the Fourier expansion
collocation method for this purpose. Following Nagaya, Paul and Venkatesan [6] studied the wave propagation in an infinite
piezoelectric solid cylinder of arbitrary cross section using Fourier expansion collocation method. The longitudinal waves
inhomogeneous anisotropic cylindrical bars immersed in a fluid are studied by Dayal [7]. Guided waves in a transversely
isotropic cylinder immersed in a fluid are analyzed by Ahmad [8]. Following Ahmad, Nagay [9] have studied the longitudinal
guided wave propagation in a transversely isotropic rod immersed in fluid, later, Nagy with Nayfeh [10] discussed the
viscosity-induced attenuation of longitudinal guided waves in fluid-loaded rods. Easwaran and Munjal [11] reported a note on
the effect of wall compliance on lowest-order mode propagation in fluid-filled/submerged impedance tubes. Sinha et. al. [12]
have discussed the axisymmetric wave propagation in circular cylindrical shell immersed in fluid, in two parts. In Part I, the
theoretical analyses of the propagating modes are discussed and in Part II, the axisymmetric modes excluding torsional modes
are obtained theoretically and experimentally and are compared. Berlinear and Solecki [13] have studied the wave propagation in
fluid loaded transversely isotropic cylinder. In that paper, Part I consists of the analytical formulation of the frequency equation
of the coupled system consisting of the cylinder with inner and outer fluid and Part II gives the numerical results. Venkatesan and
Ponnusamy [14] have obtained the frequency equation of the free vibration of a solid cylinder of arbitrary cross section
immersed in a fluid using Fourier expansion collocation method. Ponnusamy and Selvamani [15] discussed the dispersion
analysis of a generalized magneto thermo elastic cylindrical panel by introducing the thermal relaxation times. Later, Ponnusamy
and Selvamani [16] investigated the effect of rotation in an axisymmetric vibration of a transversely isotropic solid bar immersed
in an inviscid fluid using Bessel function solutions. Gladwell and Tahbildar [17] and Buchanan and Yii [18] used the finite
element method to study the 3D vibration of cylinders while Wang and Williams [19] compared their finite element result with
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Frontiers of Engineering Mechanics Research
May 2013, Vol. 2 Iss. 2, PP. 42-50
the experimental data. Singal and Williams [20] used simple algebraic polynomials as admissible functions to study the 3D
vibration of completely free hollow cylinders and the theoretical results agreed well with the experimental results. The same
problem was also studied by So and Leissa [21] using simple algebraic polynomials as admissible functions, and high accurate
results were given. Later, Zhou [22] studied the three dimensional vibration analysis of solid and hollow circular cylinders using
Chebyshev-Ritz method.
In this paper, the longitudinal waves in a homogeneous transversely isotropic rotating solid cylinder immersed in an inviscid
fluid are studied using the linearized, three-dimensional theory of elasticity. Two displacement potential functions are introduced
to uncouple the equations of motion and the solutions are obtained by Chebyshev method. The computed non-dimensional
frequencies are presented in the form of dispersion curves for the material Zinc.
II.
FORMULATION OF THE PROBLEM
We consider a transversely isotropic solid cylinder of length L and the radius ‘a’ immersed in inviscid fluid. The system is
assumed to be linear so that the linearized three-dimensional stress equations of motion are used for both the cylinder and the
fluid. The system displacements and stresses are defined by the cylindrical coordinates r ,  and z . In cylindrical coordinates,
the three-dimensional stress equations of motion and strain-displacement relations in the absence of body force are given as
 rr ,r  r 1 r ,   rz, z  r 1  rr   

 

  ( ( u)  2 u,t )  u,tt
 r ,r  r 1  ,    z , z  2r 1 r   v,tt
 rz,r  r 1 z ,   zz , z  r 1 r

 

(1a)
(1b)
  ( ( u)  2 u,t )   w,tt
(1c)
 rr  c11err  c12 e  c13ezz
(2a)
   c12 err  c11e  c13ezz
(2b)
 zz  c13err  c13e  c33ezz
(2c)
 r  2c66 er ,   z  2c44 e z ,  rz  2c44 erz
(2d)
The stress strain relation is given by
where  rr ,  zz ,  rz are the stress components, err , e , ezz , er , e z , erz are the strain components, c11 , c12 , c13 , c33 ,
c44 and
c66   c11  c12  2 are the five independent elastic constants,  is the mass density of the material. The displacement equation of



motion has the additional terms with a time dependent centripetal acceleration   (  u ) and 2  u ,t where, u  (u , 0, w) is

the displacement vector and   (0, , 0) is a constant, the comma notation used in the subscript denotes the partial
differentiation with respect to the variables.
The strain eij are related to the displacements are given by
err  u, r
e  r 1  u  v,  , ezz  w, z ,
2er  v, r  r 1  v  u,  , 2ezr   u, z  w, r  , 2e z   u , z  r 1u z , 
(3a)
(3b)
in which u , and w are the displacement components along radial, axial directions respectively. The comma in the subscripts
denotes the partial differentiation with respect to the variables. Substituting the Eqs. (3) and (2) in the Eq. (1), results in the
following three-dimensional displacement equations of motion:
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May 2013, Vol. 2 Iss. 2, PP. 42-50
c11  u,rr  r 1u,r  r 2u   r 2  c11  c66  v,
 r 2 c66u,  c44u, zz   c44  c13  w, rz
r
1
(4a)
 c66  c12  v,r     u  2w,t    u,tt
2
r 1  c12  c66  u, r  r 2  c66  c11  u,
c66  v, rr  r 1v, r  r 2 v   r 2 c11v,
c44 v, zz  r
1
(4b)
 c44  c13  w, z   v,tt
c44  w, rr  r 1 w, r  r 2 w,   r 1  c44  c13   u, z  v, z 
(4c)
  c44  c13  u, rz  c33 w, zz     2 w  2u,t    w,tt
For longitudinal wave, it is assumed that the displacement along the hoop direction, v is zero and Eqs. (4) reduces to
c11  u, rr  r 1u, r  r 2u   c44u, zz   c44  c13  w, rz
(5a)
   2u  2w,t    u,tt
c44  w, rr  r 1 w, r   r 1  c44  c13   u, z    c44  c13  u, rz
(5b)
 c33 w, zz      2 w  2u,t    w,tt
In case of real fluid there is a friction between the particles which will manifest itself in the form of equal and opposite
tangential or shearing forces on each particle at the common boundary. But in an inviscid fluid-solid interface, the perfect-slip
boundary condition allows discontinuity in planar displacement components. That is, the radial component of displacement of
the fluid and solid must be equal and the longitudinal components are discontinuous at the interface. The above coupled partial
differential equations are also subjected to the following non-dimensional boundary conditions at the surfaces r  a .

rr
 p f    rz    u  u f   0
(6)
III. SOLUTION TO SOLID MEDIUM
The Eq. (5) is coupled partial differential equations of the three displacement components. This system of equations can
be uncoupled by eliminating two of the three displacement components through two of the three equations, but this result in
partial differential equations of fourth order. To uncouple the Eq. (5), we follow Mirsky [2] and assuming the solution of Eq.
(5) as follows:

 
u (r , z, t )    r , z

,r
  ei ( t )


(7a)
 
w(r , z , t )   i a  W r , z  ei (t )


 
(7b)
 
where i  1 ,  is the angular frequency,  r , z , W r , z are the displacement potentials and
a is the geometrical
parameter of the cylinder. By introducing the dimensionless quantities such as,
 2   2 a 2 c44 , c11  c11 c44 , r  2 r a , z  2 z / H  1 c13  c13 c44 , v2  c44  , c33  c33 c44 ,  
,
 2 R 2
2  c11
T  t c44  / a
x  r / a and substituting Eq. (7) in Eq. (5), we obtain
c
 

 2   2     1  c13 W  0
11
1  c13     2   2   W  0
where  2   2 x 2  x 1  x
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(8a)
(8b)
Frontiers of Engineering Mechanics Research
May 2013, Vol. 2 Iss. 2, PP. 42-50
A non-trivial solution of the algebraic systems (7) exists only when the determinant of Eqs. (7) is equal to zero
 c        1  c   ,W  0
 
1  c 
       
2
2
11
13
2
(9)
2
13
Eq. (9), on simplification reduces to the following differential equation:
 A
4
 B 2  C    0
(10)
where


A  c11 , B   1  c11  2     , C   2   


 
2
(11)
 
Solving the Eq.(10), the displacement amplitude functions  r , z and w r , z are obtained by the orthogonal series of
Chebyshev polynomials multiplied by a geometry boundary function for the longitudinal mode as follows:
 
   A P  r P  z 
 
   d
 r , z  Fu z
2
ij
i 1 j 1
2
W r , z  Fw z
2
i
 
2
i 1 j 1
(12a)
i
(12b)
Aij Pi r Pj z
ij
The constants d ij defined in the Eq. (12b) can be calculated from the equation


d ij  1  c13  i a 
2
  a 
2
i

  2 , i  1, 2
(13)

 
are the coefficients of the polynomial. P    s  1, 2,3, 4,5 ;  r , z  is the
Where the boundary functions for the stress free boundary condition is taken Fu z  Fu1 z Fu1 z =1 unity, and i, j are the
order of Chebyshev polynomial series, Aij and Bkl
1D s
th
s
Chebyshev polynomial which can be written in terms of cosine function as follows:
Ps    cos  s  1 arc cos    s  1, 2,3...,  r , z
(14)
IV. SOLUTION OF FLUID MEDIUM
In cylindrical polar coordinates r ,  and z , the acoustic pressure and radial displacement equations of motion for an in
viscid fluid are of the form Berliner [13]
p f   B f  u, fr  r 1 (u f )  w, fz 
(15)
c f 2u,ttf  ,r
(16)
and
respectively where B f , is the adiabatic bulk modulus,  f is the density, c f  B f  f is the acoustic phase velocity in the
fluid, and  u f , w f
 is the displacement vector.
   u, fr  r 1 (u f )  w, fz 
(17)
u f  , rf and w f  , zf
(18)
 f (r ,  , z , t )   f  r   e it ,
(19)
Substituting
and seeking the solution of (15) in the form
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May 2013, Vol. 2 Iss. 2, PP. 42-50
the fluid represents the oscillatory waves propagating away is given by
 f  A33 H n ( ax)
(20)
Where  a    2  B ,
2
in which 
f
f
f
f
   f , B  B f c 44 , H n is the Hankel function of the first kind. If (2 a)2  0 , then the Hankel function of first
kind is to be replaced by K n  , where K n is the modified Bessel function of the second kind. The velocity and density relation
between the cylinder and fluid is defined by c0  v2 c f and  0    f , respectively. By substituting Eq.(18) in (15) along
with (19) and (20), the acoustic pressure for the fluid can be expressed as
p f  A33 2  H n ( ax)e 
i  Ta 
V.
(21)
FREQUENCY EQUATIONS
In this section we shall derive the frequency equation for the three dimensional vibration of the solid cylinder immersed in
fluid subjected to perfect slip boundary conditions at r  a . Substituting the expressions in Eqs. (1)- (4) into Eqs. (6), we can get
the frequency equation for free vibration as follows:
Aij  0
i, j  1, 2,3
(22)
    
    
 
    
    
 
d


 Fu  z  dr p1 r p1 z  Fu  z  p1 r p1 z 

A11  2c66 
 d F z p r d p z

 11 w   1

1
dz


d


 Fu  z  dr p1 r p2 z  Fu  z  p1 r p2 z 

A12  2c66 
 d F z p r d p z

 12 w   1

2
dz


A13   2  (ax)2  nH n ( ax)  ( ax) H n 1 ( ax) 

   p  z   F  z  p  r  drd  p  z  

   p  z   F  z  p  r  drd  p  z   E
d

E21  2   Fw z d 21
p2 r
dr

d

E22  2   Fw z d 22
p2 r
dr

1
2
u
u
2
2
1
2
23
0
E31  nJ n (1ax )  (1ax ) J n 1 (1ax )
E32  nJ n ( 2 ax )  ( 2 ax ) J n 1 ( 2 ax )
E33  nH n ( ax )  ( ax ) H n 1 ( ax )
VI. NUMERICAL RESULTS AND DISCUSSION
The coupled free waves propagation in a homogenous transversely isotropic solid cylinder immersed in water is numerically
solved for the Zinc material. The material properties of Zinc are given as follows and for the purpose of numerical computation
the liquid is taken as water.
c11  1.628  1011 Nm 2 , c12  0.362  1011 Nm 2 , c13  0.508  1011 Nm 2 , c33  0.627  1011 Nm 2 , c44  0.385  1011 Nm 2 and
density   7.14  103 kg m 3 and for the fluid the density  f  1000kg m 3 and phase velocity c f  1500ms 1 .
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May 2013, Vol. 2 Iss. 2, PP. 42-50
A comparison is made for the non-dimensional frequencies in case of free and immersed cylinder with respect to the velocity
ratio for the symmetric and anti symmetric modes of the cylinder immersed in fluid in Tables I and II. From these tables, it is clear
that as the sequential number of the velocity ratio increases, the non-dimensional frequencies also increase for both the free and
immersed cylinder. When a solid medium such as the solid cylinder is surrounded by fluid medium, guided waves are transmitted
across the interface. Thus bulk waves are excited in the embedding medium, radiating away from the solid medium.
TABLE 1 THE NON-DIMENSIONAL FREQUENCIES FOR FIRST THREE SYMMETRIC LONGITUDINAL MODES OF
THE FREE AND IMMERSED CYLINDER WITH VELOCITY RATIO
Velocity
ratio(c0)
Free cylinder
S1
S2
Immersed cylinder
S3
S1
S2
S3
0.1
0.0645
0.1236
0.1581
0.0259
0.1039
0.1284
0.2
0.1831
0.2636
0.2747
0.1742
0.1213
0.2630
0.4
0.2263
0.3928
0.4592
0.1950
0.3563
0.4220
0.6
0.3967
0.4727
0.5391
0.2837
0.4712
0.5295
0.8
0.5210
0.6036
0.7853
0.5029
0.5131
0.6752
TABLE II THE NON-DIMENSIONAL FREQUENCIES FOR FIRST THREE ANTI SYMMETRIC LONGITUDINAL MODES OF
THE FREE AND IMMERSED CYLINDER WITH VELOCITY RATIO
Free cylinder
Immersed cylinder
Velocity
ratio(c0)
S1
S2
S3
S1
S2
S3
0.1
0.0239
0.0865
0.1572
0.0215
0.0442
0.1438
0.2
0.0751
0.2319
0.2345
0.0454
0.1969
0.2205
0.4
0.1774
0.4977
0.5337
0.1444
0.3888
0.4473
0.6
0.1894
0.5485
0.7292
0.2387
0.5023
0.5951
0.8
0.3964
0.6952
0.9408
0.5504
0.6150
0.7503
VII.
DISPERSION CURVES
The result of the frequency analysis for three longitudinal modes of vibrations is plotted in the following figures with respect
to length to radius ratio of the cylinder. The notations LM1, LM2 and LM3 denotes the first, second and third longitudinal modes
respectively.
The variation of frequencies with the length to radius ratio of the cylinder is discussed in Fig.1 and Fig.2 for the free solid
cylinder with the rotational speed   0 and   0.5 for the first three longitudinal modes of vibration. In Fig.1 the frequency
increases monotonically in all the ranges length to radius ratio.
Fig. 1 Variation of frequency with L/a of free solid cylinder with
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0
Frontiers of Engineering Mechanics Research
May 2013, Vol. 2 Iss. 2, PP. 42-50
For the case of rotational parameter   0.5 there is a small deviation on the frequency pattern in Fig.2 due to the extra
driving force exerted by the rotation of the cylinder. The energy transmission occurs only on the surface of the cylinder because
the cylinder acts as the semi infinite medium. In general, from the Figs.1 and 2 it is observed that the non-dimensional frequency
of the fundamental mode is non-dispersive and increases rapidly in the presence of rotational speed with increasing wave
number.
Fig. 2 Variation of frequency with L/a of free solid cylinder with
  0.5.
In Fig. 3, the variation of frequency with respect to length to radius ratio of the solid cylinder immersed in fluid is presented
for   0 . The magnitude of the frequency increases monotonically in the range 0  L a  0.5 for first three modes of the solid
cylinder immersed in fluid and become dispersive in the remaining range of L a ratios.
Fig. 3 Variation of frequency with L/a of immersed solid cylinder with
0
The variation of frequency with respect to the length to radius ratio of the cylinder for first three longitudinal modes of the
immersed solid cylinder with   0.5. is presented in Fig. 4, where the non dimensional frequency rises slow and steady in
0  L a  0.4 and become dispersive due to the combined effect of fluid medium and rotational speed.
From Figs. 3 and 4, it is clear that the effects of length to radius ratio of the solid cylinder are quite pertinent due to the
combined effect of rotation and damping effect of the fluid medium. When the ratio of the densities of the fluid and elastic
material is small (0.14), then the mode spectrum of fluid loaded cylinder is slightly different from that of free cylinder.
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May 2013, Vol. 2 Iss. 2, PP. 42-50
Fig. 4 Variation of frequency with L/a of immersed solid cylinder with
  0.5
VIII. CONCLUSIONS
In this paper, the longitudinal vibration in a rotating homogeneous transversely isotropic solid cylinder immersed in an
inviscid fluid is studied using the linearized, three-dimensional theory of elasticity. Two displacement potential functions are
introduced to uncouple the equations of motion. In the present analysis, a set of Chebyshev polynomials multiplied by a
boundary function which satisfies the geometric boundary conditions of the cylinder are taken as the trial functions. The
computed non-dimensional frequencies for different parameters are presented in the form of dispersion curves for the material
Zinc. From the tabulated values, it is clear that the effect of the fluid force on the structure is represented as added mass, which
lowers the natural frequency of the structure from that which would be measured in a vacuum. This decrease in the natural
frequency of the fluid-structure system is caused by increasing the kinetic energy of the coupled system without a corresponding
increase in strain energy.
REFERENCES
[1] D.C.Gazis, “Three-dimensional investigation of the propagation of waves in hollow circular cylinders”. I.Analytical Formulation;II
Numerical Results, J. Acoust. Soc. Am. 31, 568-578, 1959.
[2] I.Mirsky, “Wave propagation in transversely isotropic circular cylinders”. J. Acoust. Soc. Am. 36, 41-51, 1964.
[3] K.Nagaya, “Wave propagation in an infinite long bar of arbitrary cross section and with a circular cylindrical cavity”. J. Acoust. Soc. Am.
75(3), 834-841, 1984.
[4] K.Nagaya, “Wave propagation in a rod with an arbitrary shaped outer boundary and a cylindrical cavity of arbitrary shape”. J. Acoust. Soc.
Am. 77(5), 1824-1833, 1985.
[5] K.Nagaya, “Direct method on determination of eigenfrequencies of arbitrary shaped
plates”. ASME .J. Vib. Stress Rel. Des.105,
132-136, 1983.
[6] H. S. Paul, M. Venkatesan, “Wave propagation in a piezoelectric ceramic cylinder of arbitrary cross section with a circular cylindrical
cavity”. J. Acoust. Soc. Am. 85(1), 163-170, 1989.
[7] Vinay Dayal, “Longitudional waves in a homogeneous anisotrpic cylindrical bars immersed in fluid”. J. Acoust. Soc. Am. 93(3),
1249-1255, 1993.
[8] F.Ahmad, “Giuded waves in a transversely isotropic cylinder immersed in a Fluid”. J. Acoust. Soc. Am.109(3), 886-890, 2001.
[9] B.Nagy, “Longitudinal guided wave propagation in a transversely isotropic rod Immersed in a fluid”. J. Acoust. Soc. Am. 98(1), 454-457,
1995.
[10] B.Nagy, H.Nayfeh, “Viscosity – induced attenuation of longitudinal guided Waves in a fluid loaded rods”.J. Acoust. Soc. Am. 100(3),
1501-1508, 1996.
[11] V.Easwaran, M.L. Munjal, “A note on the effect of wall compliance on lowest-order mode propagation in fluid-filled/submerged
impedance tubes”. J. Acoust. Soc. Am. 9(6), 3494-350, 1995.
[12] K.Sinha, J.Plona, S.Kostek and S. Chang, “Axisymmetric wave propagation in a fluid-loaded cylindrical shells”. I: Theory; II Theory
versus experiment, J. Acoust. Soc. Am. 92, 1132-1155, 1992.
[13] J.Berliner, R.Solecki, “Wave Propagation in a fluid-loaded, transversely isotropic cylinders”. Part I. Analytical formulation; Part II
Numerical results, J. Acoust. Soc. Am. 99, 1841-1853, 1996.
[14] M.Venkatesan, P.Ponnusamy, “Wave propagation in a solid cylinder of arbitrary cross-section immersed in a fluid, J. Acoust. Soc. Am.
(112), 936-942, 2002.
[15] P.Ponnusamy, R.Selvamani, “Dispersion analysis of a generalized magneto thermo elastic cylindrical panel”, Journal of thermal stresses.
35(12), 1119-1142, 2012.
[16] R.Selvamani, P.Ponnusamy, “Effect of rotation in an axisymmetric vibration of a transversely isotropic solid bar immersed in an inviscid
- 49 -
Frontiers of Engineering Mechanics Research
[17]
[18]
[19]
[20]
[21]
[22]
May 2013, Vol. 2 Iss. 2, PP. 42-50
fluid, Materials Physics and Mechanics, 15, 97-106, 2012.
G.M.L.Gladwell,U.C.Tahbildar, “Finite element analysis of the axis symmetric vibrations of cylinders”, J.Sound Vibr.22,143-157,1972.
G.R.Buchanan, C.B.Yii, “Effect of symmetrical boundary conditions on the vibration of thick hollow cylinders”, Appl.Acoust.63,
547-566, 2002.
H.Wang, K.Williams, “Vibrational modes of thick cylinder of finite length”, J.Sound Vibr.191, 955-971, 1996.
R.K.Singal, K.Wiliiams, “A theoretical and experimental study of vibrations of thick circular cylindrical shells and rings”, J.Vibr, Acoust.
Stress Reliab.Des.110, 533-537, 1988.
J.So, A.W.Leissa, “Free vibration of thick hollow circular cylinders from three dimensional analysis”, J.Vibr, Acoust.119, 89-95, 1997.
D.Zhou et al, “3D vibration analysis of solid and hollow circular cylinders via Chybyshev-Ritz method”, Comp. Mthods in Appl..
Mech&Eng.192, 1575-1589, 2003.
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