On Vibration of Three-Layered Cylindrical Shell with Functionally Graded Middle Layer
Zermina Gull Bhutta1, M. N. Naeem2, M. Imran2
1Department of Mathematics, University of Sargodha, Women Sub-Campus, Faisalabad, Pakistan
2Department of Mathematics, Government College University, Faisalabad, Pakistan
To cite this article:
Zermina Gull Bhutta, M. N. Naeem, M. Imran. On Vibration of Three-Layered Cylindrical Shell with Functionally Graded Middle Layer. American Journal of Applied Mathematics. Special Issue: Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015). Vol. 3, No. 3-1, 2015, pp. 32-40. doi: 10.11648/j.ajam.s.2015030301.16
Abstract: In the current analysis vibration characteristics of a cylindrical shell composed of three layers are examined. Vibration of cylindrical shells is accomplished for their involvement in various areas of engineering and technology. Shell vibration behavior depends upon on different geometrical material parameters and material parameters. They provide the maximum stability of a physical system. There is graduation distribution of constituent materials in functionally graded materials and is controlled by polynomial, exponential and trigonometric volume exponent fraction laws. In the present study a cylindrical shell is composed of three layers whereas the middle layer consists of functionally graded material and the extreme layer are of isotropic nature. Material composition of the FG layer is governed by polynomial, exponential and trigonometric volume fraction exponent laws. Impact of these laws is examined on shell vibration frequencies for different physical parameters. Love’s thin shell theory is adopted for shell motion equations. The vibration of cylindrical shells with FGM will be expressed by using the Raleigh-Ritz technique in this method. Three volume fraction laws are used to define the middle layer of tri-layer cylindrical shells. The Rayleigh-Ritz technique is applied to form the shell frequency equation which is solved by MATLAB software. The validity and accuracy of this method is investigated for a number of comparisons of numerical results.
Keywords: Component, Formatting, Style, Styling, Insert
Cylindrical shells are essential components in the field of technology as well as that of engineering. Vibrations of cylindrical shells have been extensively studied for their simple geometrical designing. So a huge amount of research on them is seen in open literature. Egle et al.  examined free vibrations of orthogonally inflexible cylindrical shells where rigidness has been treated as distinct elements. Sharma et al.  investigated vibrations of cylindrical shells for clamped-free boundary conditions by using Rayleigh-Ritz technique. The vibration of cylindrical shells with intermediary supports was examined by Swaddiwudhpong et al. . Vibrations of functionally graded (FG) cylindrical shells were investigated by Loy et al. and Pardhan et al.  for various physical parameters and several boundary conditions. Li et al.  examined vibrations of circular cylindrical shells with FG materials middle layer for simply supported end conditions. They also used Love’s approximation for strain and curvature-displacement relationships for shells The idea of tri layered cylindrical shells with intermediate layer of FGM was given by Batra  for studying axial buckling of cylindrical shells and they investigated this aspect of dynamical study of the shells. Bing et al.  examined vibration frequencies of thin walled cylindrical shells for different edge condition. Shao and Ma  investigate the vibration analysis of those cylindrical shells split into thin layer and used Fourier series expression method for SS-SS, C-C, C-F and C-SS boundary conditions. Naeem et al.  employed the Ritz formulation to investigate vibration of natural frequency characteristic of FG cylindrical shells. Naeem et al.  established the equation of FGM shells in eigenvalue expression to observe their frequencies.
In this study vibration characteristics of three layered cylindrical shell with FG middle layer are investigated. The frequencies analysis of two layers cylindrical shells was examined by Arshad et al.  in which one layer was FG layer and other layer was of homogeneous materials. Iqbal et al.  examined vibrations of FG cylindrical shells applying the wave propagation technique. The generalized differential quadrature method was applied to examine the vibration characteristics of FG materials cylindrical shells by Naeem et al. . Sofiyev et al.  examined the non-linear free vibration of FG cylindrical shells attached to combine loads with various ends conditions and resting on elastic foundations. Vel  employed the elasticity solution technique to observe free and forced vibration of cylindrical shells. These shells were estimated by SS-SS boundary condition. Shah et al.  applied exponential volume fraction law to observe the cylindrical shell’s vibration with FGM. Warburton et al.  investigated the appearance of frequency variations with the circuit wave and expressed the frequency in the form of shell energies. Vibration of spinning cylindrical shells was examined by Mehparvar . The shells were constructed from FGM. They used the higher ordered theory for shell deformation with the use of energy Hamilton’s principle to obtain the shell dynamical equations. The vibration of cylindrical shells which are containing FGM was observed by Lam et al. . Their purpose was to check the effect of FGM on vibration characteristics of the shells. Their composition was maintained by volume fraction power law of distribution of materials in the radial direction. Yamanouchi et al. and Koizumi  studied the structure and design of FGMs.
In this paper vibration of three layered cylindrical shells are analyzed for various shell parameters. The shell thickness consists of three layers where materials of the outer layers are of isotropic. The middle layer consists of FG materials. The shell problem has been written in the integral form by considering expressions of kinetic and strain energies for a cylindrical shell. The shell frequency equation is formed by applying the Raleigh-Ritz technique. The estimation of axial modal dependence is done by characteristic beam functions. These functions satisfy boundary conditions. Results are obtained for simply supported- simply supported, clamped-clamped, clamped- free and clamped-simply supported boundary conditions. Comparisons of results determined by this procedure are done with those found in literature to verify the validity and efficiency of this technique and accuracy of the results.
2. Theoretical Formulation
Figure 1, represents the geometry of a cylindrical shell., stand for its geometrical quantities viz.; length, thickness and mean radius respectively while designate Young’s modulus, the Poisson ratio and the mass density respectively. The triplet defines an orthogonal coordinate system and they lie at the mid plane of the cylindrical shell. They describe the coordinates in the longitudinal, tangential and transverse directions respectively. The functions and indicate for the longitudinal, tangential and transverse displacements from the mid surface of the shell.
For a vibrating thin cylindrical shell, its strain energy, expressed by U is stated as Loy et al. :
where the stress where and define the reference surface strains, ,and represent the surface curvatures and whereand and are associated with the extensional , coupling and bending stiffness respectively and are stated as :
The reduced material stiffness andfor isotropic materials are described as ::
for isotropic cylindrical shells the coupling stiffness considered equal zero and for the shells formed by FGM they considered non-zero. For the cylindrical shells which fabricated by FG materials their values depend on the material distribution. The negativity and positivity of coupling stiffness exist due to the irregularity of characteristics of materials at mid plane when reduced stiffness produced by physical properties of FG materials.
Also the kinetic energy of the cylindrical shell, denoted by , is written as 
where t denotes the time variable and represents the mass density per unit length and is written as
wherestand for the mass density.
2.1. Love’s Shell Theory
Several shell theories have been found in the open literature. Kirchhoff’s assumption is the basis for all shell theories. This assumption states that "Normal to the original mid-surface of a shell retains its normal position, suffer no change in length during deformation". Shell theory due Love is the pioneering one and all other modern theories have designed from it by modifying some physical terms. The formulas for strain and curvature–displacements are adopted from Love’s shell theory to solve the present shell problem and are written as:
These expressions for the surface strains and and the curvatures and from the relations (6) and (7) respectively are replaced into Equ.(1), the expression for strain energy, attains the following form:
The Lagrange energy functional, symbolized by for a cylindrical shell is described by the difference of its strain and kinetic energies as:
The Raleigh-Ritz technique is used to examine the vibration of cylindrical shells. The deformation of cylindrical shells in longitudinal, tangential and transverse direction describe in the form of shell motion’s equations with particular variables. Many kinds of mathematical functions are used to measure the axial modal dependence. The boundaries conditions of cylindrical shells are satisfied by them.
2.2. Modal Displacement Functions
The unidentified displacement functions and showing deformations in the longitudinal, tangential and transverse directions are supposed in such shapes that the separation of the special and temporal variables is performed. This process is done by classical technique of separation of variables used for solving partial differential equations. The substitution of the presumed shapes of the modal displacement functions are made into the shell governing equations and a system of simultaneous equations is obtained in the vibration amplitude coefficient by the Rayleigh-Ritz method. The axial modal dependence related to the unknown functions is used to determine those functions which meet boundary conditions described for cylindrical shells. The following models for the modal deformation function are mentioned for axial, tangential and temporal variables:
anddenotes the frequency of the cylindrical shell and n is the circumferential wave number. The coefficients A, B, C show the vibration amplitudes in the longitudinal, tangential and transverse directions respectively.
Substituting the above expressions of the shell energies into Equation (9), the new expression for the Lagrange functional is achieved as
Applying the Rayleigh- Ritz method, the process of minimization is applied to the Lagrange functional and is partially differentiated with regard to the vibration amplitude coefficients A, B and C. So doing process of extremization of, the following required minimum value conditions are obtained:
2.3. Derivation of the Shell Frequency Equation
The point when terms of these compelling conditions adjusted in particular shape then shell recurrence mathematical statement is discovered. Three concurrent mathematical statements in A, B, C are acquired as:
where the coefficients are some constants. The above equations can be written in the matrix form as
This represents the frequency equation in the eigenvalue problem form. The condition of making the determinant of the matrix coefficients zero is applied for non-trivial solution for achieving the frequency equation.
2.4. Polynomial Volume Fraction Law
The properties of FG materials vary for temperature and they are originating in the field of high thermal condition. If the material property is denoted by P which is function of the absolute temperature T(K). Then Touloukian (1973) stated as:
where the thermal coefficients are indicated by P0, P-1, P1, P2 and P3 while T indicates the temperature at absolute scale. The material properties of a FG constituent material for a cylindrical shell are functions of both temperature and their volume fractions. The succeeding material of a FG material is described as:
where the materials characteristics are mentioned by and the volume fraction of FGM denoted by. Their sum always equal to one
denotes the volume fraction of a FG material. It can be written as:
The thickness of cylindrical shell denoted by h and power-law exponent by N and its value always lie between zero and infinity. FGM are composition of two materials. For a FG cylindrical shell, are expressed as:
where denotes the materials used for M2 and describe the materials for. Both the materials present on the inward and outward surfaces of cylindrical shells can change their materials characteristics by interchanging themselves. The cylindrical shells with FGM are usually in-homogeneous shell. When the thickness of a shell toward its radius ratio is less than 0.05 then the theory of classical thin-walled cylindrical shell is applicable.
2.5. Exponential Volume Fraction Law
Arshad et al.  modified the polynomial volume fraction law (20) and framed it in the exponential expression as:
where … is the usual natural base. Further formula is amended and a more general base is established and a new expression is written as:
Thus formulae for the effectual material properties: the effective Young’s modulus the Poisson ratio and the mass density for a FG are written as:
The above relations express M2 present at the inward surface while M1 at outward surface of the cylindrical shells.
2.6. Trigonometric Volume Fraction Law
This law obtained by making some changing in the formulae defines in (20) and (25) for cylindrical shell with FG layer related to and can be defined as:
where is a positive real number. The conclude materials for this law can also express like other two laws for cylindrical shells with FG
From formulae (30), when and when Thus at
M2 is attached at inward side but when z=h/2 then the characteristics material are obtained by both M1 and M2 materials present at outward surface of the cylindrical composed with FG material.
2.7. Material Stiffness for Three-Layered Cylindrical Shells
The thickness layer of the cylindrical shell is divided into three layers. Thicknesses of interior, intermediate and exterior layers are h1, h2 and h3 respectively. For simplicity, thickness of each layer is of the thickness h/3. According to this configuration, the coefficients of extensional, coupling and bending stiffness Aij, , Bij and Dij are modified as
Here E, E2 and E2 are Young’s moduli, N is power-law exponent and vf volume fractions while v,v1 and v2 are Poisson ratios.
3. Result and Discussion
The comparison of values of non-dimensional frequency parameters, for simply supported boundary conditions for homogeneous cylindrical shell with those of Loy et al.  is composed in Table 1. The present case was solved by the Raleigh-Ritz method while the frequency parameters in Loy et.al.  were obtained by the differential quadrature method. This comparison shows that the present results are nearly equal with each other. At n=2, the frequency parameter has the lowest value.
Table 1. Comparison of frequency parameters for a cylindrical shell with simply supported;-simply supported boundary conditions .
|n||Loy et al. ||Present|
Table 2. Comparison of natural frequencies (Hz) for a simply supported- simply supported isotropic cylindrical shell (L=8in, h=0.1in, v=0.3, =7.35×Ibfs2 in-4, E=30×106Ibf in-2).
A comparison of the result of natural frequencies (Hz) for a cylindrical shell for simply supported-simply supported edge conditions is given with the results of Warburton  in the Table 2. These boundary conditions are applied at the both end points of the cylindrical shell. The half-wave axial numbers are taken to be m = 1, 2, 3, 4, 5, 6 and the circumferential wave numbers are taken n=2, 3. From the comparison it observed that these results are close to each other.
The results frequencies (Hz) of vibration cylindrical shells having FGM are obtained. These cylindrical shells consisof two types of FG material. Two materials: nickel and stainless steel are associated at inward and outward surfaces of a FG cylindrical shell of 1st Type. While in 2nd Type they interchange their positions. The outer surface denoted by M1 and inner denoted by M2. Natural frequencies (Hz) of 1st Type and 2nd Type cylindrical shells are composed in Table 3 and 4 respectively for the half-axial wave mode m =1. Geometric parameters are mentioned in the Tables. Polynomial fraction law regulates the material distributions in FGM. The power law exponents are taken as: N= 0.5, 1, 15.The present obtained frequencies and those of Iqbal et al.  are compared with each other. The shell frequencies have been evaluated by the Raleigh - Ritz method and wave propagation method was applied by Iqbal et al.  to obtain them. The condition which is stated at both the ends is simply supported-simply supported. So the compared results coincided with each other.
|Iqbal et al. ||Present|
|Iqbal et al. ||Present|
From the above comparisons, it is clear that the present numerical procedure is efficient and valid and yields accurate results.
Natural frequencies (Hz) for the present configurations of three layered cylindrical shells are furnished with variations depending on circumferential wave number, n for the half axial wave numbers, considering m=1. The end conditions considered here are simply supported – simply supported (SS-SS), clamped-clamped (C-C), clamped- free (C-F) and clamped- simply supported (C-SS). The three volume fraction laws: (i.) polynomial, (ii.) exponential and (iii.) trigonometric are applied to measure the material composition of FG layer.
From the Tables 5 and 6, it is observed the natural frequencies (Hz) of 1st Type of cylindrical shells with four boundary conditions like SS-SS, C-C, C-F and C-SS for polynomial volume fraction law decreases when the value of power exponent N increases
The Tables 7 and 8, describe the natural Frequencies of 2nd Type of cylindrical shells with four boundary conditions like SS-SS, C-C, C-F and C-SS for polynomial volume fraction law. It is observed the frequencies increase when power exponent N increases
The variations of cylindrical shells having FG middle layer with SS-SS, C-C, C-F and C-SS boundary conditions for both types described in the Tables 9-12 for versus n for the half- axial wave mode, m=1 with exponential fraction law.
It is observed from the Tables 8 and 9, the natural frequencies (Hz) of 1st Type of cylindrical shells with SS-SS, C-C, C-F and C-SS boundary conditions for exponential volume fraction law decreases when the value of power exponent N increases.
The natural frequencies (Hz) of 2nd Type of cylindrical shells with same mentioned above boundary conditions and volume fraction law catalogued in the Tables 11 and 12. It is observed the behaviour of natural frequencies is reverse of 2nd Type.
The variations of cylindrical shells having FG middle layer with SS-SS, C-C, C-F and C-SS boundary conditions for both types described in the Tables 13-16 for versus n for the half- axial wave mode, m=1 with trigonometric fraction law.
It is observed from the Tables 13 and 14, the natural frequencies (Hz) of 1st Type of cylindrical shells with SS-SS, C-C, C-F and C-SS boundary conditions for trigonometric volume fraction law decreases when the value of power exponent N increases.
The natural frequencies (Hz) of 2nd Type of cylindrical shells with same mentioned above boundary conditions and volume fraction law catalogued in the Tables 15 and 16. It is observed the behaviour of natural frequencies is reverse of 1st Type.
The vibration of cylindrical shells with FGM express by using the Raleigh-Ritz technique in this method. Three volume fraction laws are used to define the middle layer of tri-layer cylindrical shells. Two types of cylindrical shells are discussed in this method. The middle layer of cylindrical shell is FG which is composition of two materials Nickel and Stainless steel. At the inward surface of shell Stainless steel attached, while Nickel is attached at outward surface in 1st Type of shells. The position of these materials will interchange in 2nd Type. The results for simply-supported-simply supported, clamped-clamped, clamped-free and clamped- simply supported boundary conditions are obtained by this method. Following results are obtained by this present shell problem.
I. Circumferential wave number affect on the natural frequencies (Hz) of both Types of cylindrical shells. The frequencies increased and decreased by them.
II. Comparison of present obtained results with exponent power law for three volume fraction laws with the results of Loy et al. and Naeem et al. [10-11] shows that they are good agreement with each other.
III. It observe that in 1st Type of cylindrical shell frequency is increasing as N increase and in 2nd Type it decreasing when N increase, due to interchanging the materials M1 and M2.
IV. The comparison of frequencies values of three volume fraction laws give the result that the frequency of 1st Type cylindrical shell increasing by polynomial fraction law, while in 2nd Type the frequency of cylindrical shells with clamped-clamped boundary condition increased by exponential law and other with polynomial fraction law. The comparison of variations of frequencies estimated that the recent method is valid and accurate. The obtained results are very close to previous result.