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Free Vibration Analysis of Embedded Circular Cylindrical Shells Using Wave Propagation Approach

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INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Vol.1 Issue.7, Novembe !"1#. $%s& '()(*
T. Srinivas, V. V. Sridhara Raju, S. Narendar*
'(
ISSN (ONLINE): 2321-3051
INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Free Vibration Analysis of Embedded Circular Cylindrical Shells Using Wave Propagation Approach
T. Srinivas
1
, V. V. Sridhara Raju
1
, S. Narendar
2,*
1
Department of Mechanical Engineering, PVP Siddhartha Institute of Technology, Kanuru, Vijayawada 520 007, India, E-mail: teege.srinivas@gmail.com;
vvsridhrarraju@yahoo.co.in
2
Defence Research and Development Laboratory, Hyderabad 500 058, India. Author Correspondence: nanduslns07@gmail.com , Tel: +91 8897 625977
Abstract
The present research work deals with the study of the free vibration of buried/embedded shells. These buried shells are modelled as circular cylindrical shells as embedded in elastic medium. The elastic medium is approximated as Pasternak model. The bonds are assumed to be formed between the shell and the elastic medium. The elastic matrix is described by a Pasternak foundation model, which accounts for both normal pressure and the transverse shear deformation of the surrounding elastic medium. When the shear effects are neglected, the model reduces to Winkler foundation model. The normal pressure or Winkler elastic foundation parameter is approximated as a series of closely spaced, mutually independent, vertical linear elastic springs where the foundation modulus is assumed equivalent to stiffness of the springs. Governing equations for this system are derived from variations principles and the solution to the vibration problem is done by using wave propagation method. The obtained natural frequencies are presented by using the symbolic tool box of the MatLab
®
software. The effects of the surrounding matrix, axial and circumferential modenumbers, length of the shell, thickness of the shell and the various boundary conditions are presented and discussed in detail.
Keywords
:
Vibration, Elastic Shell; Wave Propagation Method; Natural Frequency; Mode number.
1. Introduction
Shells are the fundamental structures in the field of aerospace, civil and mechanical engineering disciplines. Cylindrical shells are the practical elements of many types of engineering structures such as aeroplanes, ships and construction buildings. Machinery-induced vibration often occurs in these structures, like a pipe, it often transmits unwanted energy in the form of structural and acoustic vibrations besides transmitting useful quantities. The vibrations of cylindrical shells have been extensively investigated in the literature. A number of solution techniques have been developed for various boundary conditions as reviewed by Leissa (1973). In addition to these mechanical advantages, shell structures enjoy the unique position of having extremely high aesthetic value in various architectural designs Contemporary engineers using scientifically justified methods of design tend to develop a structure that combines maximum strength, functional perfection, and economy during its lifetime. In addition, it is important that the best engineering solution
INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Vol.1 Issue.7, Novembe !"1#. $%s& '()(*
T. Srinivas, V. V. Sridhara Raju, S. Narendar*
+"
ensues, other things being equal, at the expense of the selection of structural form and by not increasing the strength properties of the structure, e.g., by increasing its cross section. Note that the latter approach is easier. Shell structures support applied external forces efficiently by virtue of their geometrical form, i.e., spatial curvatures; as a result, shells are much stronger and stiffer than other structural. We now formulate some definitions and principles in shell theory. The term shell is applied to bodies bounded by two curved surfaces, where the distance between the surfaces is small in comparison with other body dimensions The locus of points that lie at equal distances from these two curved surfaces defines the middle surface of the shell. The length of the segment, which is perpendicular to the curved surfaces, is called the thickness of the shell and is denoted by
h
. The geometry of a shell is entirely defined by specifying the form of the middle surface and thickness of the shell at each point. In this book we consider mainly shells of a constant thickness. Shells have all the characteristics of plates, along with an additional one –curvature. The curvature could be chosen as the primary classifier of a shell because a shell’s behavior under an applied loading is primarily governed by curvature. Depending on the curvature of the surface, shells are divided into cylindrical (noncircular and circular), conical, spherical, ellipsoidal, paraboloidal, toroidal, and hyperbolic paraboloidal shells. Recently, the wave number of axisymmetric waves of buried pipes using Kennard’s equations for low frequency (below the ring frequency) as predicted by Muggleton (2002) and give some experimental results Muggleton (2004), but the papers deal the effects of outer medium on the pipes with sound pressure fields that are scalar variables. In fact, the surrounding medium exerts effects on the pipes at all direction. Loy et al., 1998, solved the vibration of cylindrical shells by the method of generalized deferential quadrature (GDQ). Chen et al., (1998) studied the dispersion characteristics of circular cylindrical shells for low and high circumferential modes. Based on the Love’s equations, Wang and Lai (2000) presented a wave approach for predicting the natural frequencies of finite length circular cylindrical shells with different boundary conditions. El-Mously (2003) performed a comparative study of three approximate ‘‘explicit’’ formulae for estimating the fundamental natural frequency of a thin cylindrical shell. Zhang et al., (2001) studied the vibration characteristics of thin cylindrical shells using wave propagation. They calculated the frequencies for a long shell (L/R = 20), and compared the results with those obtained using FEM (MSC/NASTRAN). Karczub (2006) studied the algebraic expressions to determine the structural wave numbers based on the Flügge equations of motion. Zhang and Xiang (2007) applied the exact solutions to the vibration of circular cylindrical shells with step-wise thickness variations in the axial direction. Using a wave propagation approach based on Flügge classical thin shell theory, Li (2008) investigated the free vibrations of circular cylindrical shell with classical homogeneous boundary conditions. Ma et al., (2009) developed a modal model for cylindrical shells by combining the wave propagation approach with an improved Fourier series method. The axial wavenumbers used there were obtained approximately from the beam theory. Caresta and Kessissoglou (2009) studied the low frequency vibration behavior of and sound radiation from a cylindrical shell under axial excitation, the free vibration characteristics of isotropic coupled conical–cylindrical shells Caresta and Kessissoglou (2010) and the structural and acoustic responses of a submarine hull under harmonic force excitation Caresta and Kessissoglou (2010). The shell is modelled as a thin cylindrical shell of linear homogeneous isotropic elastic material and embedded in a linear isotropic homogeneous elastic medium of infinite extent. The embedded pipe connected two fire hydrants through flanges is simplified as a simply supported pipe and the two fire hydrants are simplified two hinged ends. The focus of this paper is on the research of the free vibration of simply supported embedded shells using the wave propagation method, and this is the basis of analyzing the propagation wave speeds and wave attenuation of embedded shells. The vibrations of the shell are examined by using various shell equations. The natural frequencies are obtained for the shells surrounded by elastic medium. The stiffnesses for the restraining springs can be specified to arbitrary values and all the classical homogeneous boundary conditions can now be treated in a unified manner. Several numerical examples are presented to show the effects of the elastic restraints. The effects of the surrounding matrix, axial and circumferential modenumbers, length of the shell, thickness of the shell and the various boundary conditions are presented and discussed in detail.
INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Vol.1 Issue.7, Novembe !"1#. $%s& '()(*
T. Srinivas, V. V. Sridhara Raju, S. Narendar*
+1
2. Mathematical Formulation
The schematic of the shell model is shown in Fig. 1. The well-known strain-displacement equations of the three-dimensional theory of elasticity in orthogonal curvilinear coordinates are
( )
31
1, 1,2,3 12
iik ik iik k i
UgU eiggg
α α
=
∂∂= + =
∂ ∂ √
∑
( )
1,1,2,3 2
jiijij jiijij
U U ggijijgggg
γ α α
∂ ∂
= + = ≠
∂ ∂√
Where the
,,
iij
e
γ
and
i
U
are normal strains, shear strains, and displacement components, respectively, at an arbitrary point. In the shell coordinates the indices 1, 2, and 3 are replaced by
,,
andz
α β
, respectively, except for the displacements
123
, ,
UUandU
which are replaced by U, V, and W, respectively. Figure 1: Schematic of the shell showing the coordinates and the dimensional representation of the shell Now in order to satisfy the Kirchhoff hypothesis, the class of displacements is restricted to the following linear relationships
( ) ( ) ( ) ( )
,,,, 3
Uzuz
α
α β α β θ α β
= +
( ) ( ) ( ) ( )
,,,, 4
Vzvz
β
α β α β θ α β
= +
( ) ( ) ( )
,,, 5
Wzw
α β α β
=
Where u,v, and w are the components of displacement at the middle surface in the
,
α β
,and normal directions, respectively, and
and
α β
θ θ
are the rotations of the normal to the middle surface during deformation about the
β
and
α
axes, respectively ;i.e.,
INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Vol.1 Issue.7, Novembe !"1#. $%s& '()(*
T. Srinivas, V. V. Sridhara Raju, S. Narendar*
+!
( )( )
,, 6
Uz z
α
α β θ
∂=∂
( )( )
,, 7
Vz z
β
α β θ
∂=∂
2.1 Equations of Motion
The shell coordinates to be used are x and
θ
. Further, the length coordinate x is replaced by a nondimensional length s defined by
xs R
=
Where R is the cylindrical radius.
( ) ( )
1 81
ezk z R
α α α α
= +
+
ò
( )
( )
1 91
ezk z R
β β β β
= +
+
ò
( )
2
111 102211
zzz z RRRR zz RR
αβ αβ α β α β α β
γ τ
= − + + +
+ +
ò
2.3.1 Equations of Love and Timoshenko
If in above equations one neglects the terms
zzand RR
α β
and their products as being small in comparison with unity one obtains
( )
11
ezk
α α α
= +
ò
( )
12
ezk
β β β
= +
ò
( )
13
z
αβ αβ
γ τ
= +
ò
2.2 Force and Moment Results
Thus, if the relationships between stresses and strains are defined (as, for example, in Hooke’s Law), the resulting stress can be integrated over the shell thickness. The resultants of the integrals will be termed “force resultants” and “moment results” in this work. Other terminologies for these quantities used variously in the literature of shells include “stress resultants” and “forces”, corresponding to our force resultants, and “stress couples”, “couples”, “couple resultants”, and “moments”, corresponding to our moments. The force and moment resultants will have dimensions of force per unit length and moment per unit length, respectively.
INTERNATIONAL JOURNAL OF RESEARCH IN AERONAUTICAL AND MECHANICAL ENGINEERING
Vol.1 Issue.7, Novembe !"1#. $%s& '()(*
T. Srinivas, V. V. Sridhara Raju, S. Narendar*
+#
Hooke’s Law will first be assumed as the constitutive law to be followed. This limits all shells considered in this monograph to be made from materials which are linearly elastic. Furthermore, in this chapter devoted to deriving shell theories in their most simple forms, the materials will be limited to those which are isotropic. Hook’s Law is written in its well-known three-dimensional form as
( )
( )
1 14
z
e E
α α β
σ ν σ σ
= − +
( ) ( )
1 15
z
e E
β β α
σ ν σ σ
= − +
( )
( )
1 16
zz
e E
α β
σ ν σ σ
= − +
( )( )
21 17
E
αβ αβ
ν γ σ
+=
( )( )
21 18
zz
E
α α
ν γ σ
+=
( )( )
21 19
zz
E
β β
ν γ σ
+=
Where,
and
α β
σ σ
are the normal stresses and
and
αβ βα
σ σ
are the shear stresses in the tangential
( )
and
α β
directions and
zz
and
α β
σ σ
are the transverse (i.e., in the z direction) shear stresses, all acting upon the transverse faces of a shell element; E is Young’s modulus,
ν
is poisons ratio. Assuming the symmetry of the stress tensor (neglecting body couples), then
αβ βα
σ σ
=
. It is pointed out that the strains are also assumed to be independent of temperature also assumed to be independent of temperature because temperature has no explicit effect upon the free vibration case being considered in this monograph. The Kirchhoff hypothesis, yields
0,0
zzz
e
αβ β αβ β
γ γ σ σ
= = = = =
and
( )
.
α β
σ ν σ σ
= +
But Love’s third assumption is that
z
σ
is negligibly small, which is one unavoidable considered here.Another contradiction is that
zz
and
α β
σ σ
are clearly not zero, since their integrals must supply the transvers shearing forces needed for equilibrium; but they are usually small in comparison with
,,.
and
α β αβ
σ σ σ
Retaining the assumption that
z
σ
is negligibly small reduces the problem to one of plane stress; that is, equations reduce to
( )
1 20
e E
α α β
σ νσ
= −
( )
1 21
e E
β β α
σ νσ
= −

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