`Advances in Acoustics and VibrationVolume 2012, Article ID 507031, 11 pageshttp://dx.doi.org/10.1155/2012/507031`
Research Article

## Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations

1Government Post Graduate College Jhang, Jhang 35200, Punjab, Pakistan
3Department of Civil Engineering, Iqra National University Peshawar, Peshawar 25000, Pakistan

Received 28 April 2012; Revised 31 July 2012; Accepted 15 August 2012

Copyright © 2012 Abdul Ghafar Shah et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Vibrations of a cylindrical shell composed of three layers of different materials resting on elastic foundations are studied out. This configuration is formed by three layers of material in thickness direction where the inner and outer layers are of isotropic materials and the middle is of functionally graded material. Love shell dynamical equations are considered to describe the vibration problem. The expressions for moduli of the Winkler and Pasternak foundations are combined with the shell dynamical equations. The wave propagation approach is used to solve the present shell problem. A number of comparisons of numerical results are performed to check the validity and accuracy of the present approach.

#### 2. Shell Problem

A cylindrical shell of the thin wall is considered here as shown in Figure 1, with the geometrical parameters: length L, thickness , and mean radius . The orthogonal coordinate system is taken to be at the surface of the shell where , , and represent the axial, circumferential, and radial coordinates, respectively. Young’s modulus E, the Poisson ratio , and the mass density ρ are the shell material parameters. The axial, circumferential, and radial displacement deformations are denoted by , , and , respectively, with regard to the shell middle surface.

Figure 1: Geometry of three-layered cylindrical shell.

For a cylindrical shell, plane stress condition can be assumed. From two-dimensional Hook’s law, the constitutive relations for a thin cylindrical shell are stated mathematically as where stands for the stress vector, corresponds to the strain vector, and denotes the reduced stiffness matrix. These stress and strain column vectors are defined as where and are linear stresses in the and -directions, and is the shear stress in the -plane. and are strain in the directions and , and the shear strain in the -plane. The reduced stiffness matrix is written as for the plane stress condition as So the relation (1) can be expressed as

The reduced stiffness for isotropic materials is defined as

The strain vector in its component form is defined aswhere , , and denote the reference surface strains. , , and represent the surface curvatures.

From the relations (4), (6a), (6b), and (6c), the stress vector in its component form is defined as

The force resultants and moment resultants for a cylindrical shell are written in the integral forms, respectively, as where , , and are force components in axial, circumferential, and shear directions. , , and are moment components in axial, circumferential, and shear directions.

The relations (6a), (6b), (6c), (8), and (9) imply that where and are defined as and is defined as where ,  , and are submatrices of extensional, coupling, and boundary stiffness and are given as From , , and , a matrix can be written as where , , and stand for extensional, coupling, and bending stiffness, respectively, and are defined as By using the relations (11)–(15), the expression (10) takes the following form:

The (coupling stiffness) ’s are vanished for isotropic cylindrical shells and are nonzero in case of FGM cylindrical shells.

For a thin-walled cylindrical shell that performs vibration, the formula for its strain energy denoted by U is written as

By substitution of from (12) and (15) in the above expression, the new expression of the shell strain energy is written as Also the kinetic energy of the cylindrical shell, represented by , is expressed as where symbolizes the time variable and denotes the mass density per unit length and is defined as

The present problem is performed by the Love’s [1] first order thin shell theory and the analysis of expressions strain-displacement and curvature-displacement relationships are taken from this theory and are written as In a cylindrical shell executing vibration. The expressions for surface strains , and and the curvatures , and from the relations (22), respectively, are substituted into (19), and the expression for strain energy is obtained as

The Lagrangian energy functional represented by for a circular cylindrical shell is defined by differentiating its strain and kinetic energies as

Hamilton’s principle is employed to the Lagrangian energy functional , and introducing the Winkler and Pasternak foundations ( term in the -direction is inducted and appended with the equations of motion to derive the shell motion equation written in a differential operator form as where state the differential operators with regard to and and are given in Appendix A, where stands for Pasternak elastic foundation and for the Winkler foundation modulus. The expression for the differential operator is The Pasternak model reduces to the Winkler model when vanishes.

Using the expressions for the differential operators , the shells dynamical equations are written as where are defined in Appendix B.

#### 3. Methodology

The following modal displacement shape functions are adopted to separate the time and space variables: in the longitudinal, circumferential, and transverse directions, respectively. The constants , , and are the amplitudes of vibrations in the , , and directions, respectively, n is the number of circumferential waves, and stands for axial wave number that is associated with a boundary condition is axial wave number given for four types of boundary conditions in Table 1.

Table 1

These axial wave numbers are selected to satisfy boundary conditions at both edges of a cylindrical shell. denotes the natural angular frequency for the cylindrical shell.

#### 4. Derivation of Frequency Equation

Making substitution for the displacement functions u, , and from the expressions (28) in system of (27a), (27b), and (27c), the shell algebraic equations are written as:where are given in Appendix C.

After the arrangement of terms, the algebraic simultaneous equations (29a)–(29c) are written in matrix notation as

For forming the shell frequency equation, the determinant of the matrix coefficients is vanished for nontrivial eigenvalue of the shell frequency. Either the eigenvalue problem obtained above is solved using computer program or the determinant is expanded and the frequencies equation is achieved in the form of polynomial equation involving .

#### 5. Results and Discussion

Numerical results for simply supported-simply supported cylindrical shell are calculated and compared with those found in the open literature in order to examine validity, efficiency, and accuracy of wave propagation approach. In Table 2, the nondimensional frequency parameters for an isotropic cylindrical shell are compared with those ones obtained by Naeem and Sharma [20]. The simply supported boundary conditions are described on shell ends. In this comparison, the shell parameters: length , thickness , and , are taken, respectively. The axial mode is assumed to be , and the circumferential wave numbers are chosen from 1–10. Material properties of the shell are mass density (), Poisson’s ratio (), and Young’s modulus () and their values are given as kg/m3, , and  N/m2.

Table 2: Comparison of frequency parameter for SS-SS cylindrical shell (, , , , ).

In Table 3, the frequency parameters for a cylindrical shell with simply supported boundary conditions are compared with those results determined by Zhang et al. [8]. The geometrical parameters are assumed to be as , , and . For the parameters , a value of is used and are selected 0–4 in the comparison. The material properties of the shell are given as = 7850 kg/m3, , and  N/m2.

Table 3: Comparison of frequency parameter for SS-SS cylindrical shell (, , ).

#### 6. Three-Layered Cylindrical Shells

A number of results for the proposed three-layered functionally graded cylindrical shells are determined for various sets of material and geometrical parameters. The inner and outer layers of the shell are comprised of isotropic material, whereas the middle layer is assumed to be functionally graded as shown in Figure 1. In general, vibration characteristics are most influenced by Young’s modulus. In this study, the Poisson’s ratio is presumed to be constant for functionally graded materials, whereas the Young’s modulus depends on intrinsic thickness variable (z) as well as the Young’s modulus of constituent materials forming functionally graded layers. Here two configurations of a cylindrical shell are considered to suggest with regard to the shell layer thickness. In the first configuration, the thickness of each layer is supposed to be of while in the second configuration, the thickness of each of the inner and outer layers is of and that of the middle layer is of .

The stiffness moduli , , and are modified in according to the thickness of material layers when inner and outer layers are isotropic and middle, is functionally graded aswhere and in(isotropic), , and out(isotropic) are associated with inner isotropic, middle functionally graded and outer isotropic layers of cylindrical shell, respectively. Here, by considering the constituent materials of stainless steel and aluminum for isotropic layers and also the FG layers which are structured from two kinds of materials, nickel and zirconia four types of shells are obtained and are listed Table 4.

Table 4: Description of cylindrical shells.

Material properties of isotropic materials: steel and aluminum are given in Table 5, whereas the material properties of the constituent materials forming functionally graded layers are listed in Table 6.

Table 5: Material properties of isotropic materials.
Table 6: Material properties of Nickel and Zirconia.

The FGMs are much advanced materials and are used in engineering science and technology. Material properties of an FGM are the functions of the temperature and the volume fractions. These properties of a constituent material are managed by a volume fraction. If represents a material property of the th constituent material of an FGM consisting of constituent materials, then the effective property of the FGM is written as where is the volume fraction of the th constituent material. Also, the sum of volume fractions of the constituent materials is equal to 1, that is, The volume fraction depends on the thickness variable and is defined as for a cylindrical shell. and denote inner and outer radii of the shell, respectively, and is the thickness variable in the radial direction. is known as the power law exponent. It is a nonnegative real number and lies between zero and infinity. For a cylindrical shell, the volume fraction is assumed as where is the shell uniform thickness. When the shell is considered to consist of two materials, the effective Young’s modulus , the poisson ratio , and the mass density are given by

#### 8. Frequency Analysis of Cylindrical Shells

##### 8.1. Variations of Natural Frequencies (Hz) against Circumferential Wave Number (n)

In Table 7, natural frequencies (Hz) of four types of cylindrical shells are given against the circumferential wave numbers (n). The boundary conditions applied on the shell ends are simply supported. The shell parameter data is . For functionally graded layer, the power law exponent (p) is 5. The numerical results exhibit the well-known characteristic of shell vibration frequencies, that is, the value of the frequency first decreases achieves its lowest value and then it begins to increase with the circumferential wave modes (n).

Table 7: Variation of natural frequencies (Hz) of a three-layered simply supported-simply supported cylindrical shell (, , , , ).

For Type 1 and 2, the outer and inner layers are of steel. This shows that the values of shell frequencies increase with interchanging the order of constituent materials forming the functionally graded middle layer. For Type 1, the inner and outer layer is of steel whereas in Type 3, the inner is of steel and the outer is of aluminum. The frequency increases with the interchange of inner and outer layers material whereas the configuration of functionally graded layer is the same. Also in Type 4, the inner and outer layers possess the same materials as in Type 3, where the materials of middle layer have been replaced by changing the order of zirconia and nickel. This shows that the frequency has been increased by changing the materials of functionally graded layer.

In Tables 8 and 9, variations of natural frequencies for Type 1 for two configurations with respect to thickness of layers are shown against circumferential wave numbers (n). It is seen from Table 8 that by decreasing the thickness of the middle functionally graded layer, the frequencies diminish.

Table 8: Variation of natural frequencies (Hz) of a three-layered simply supported-simply supported cylindrical shell for two configurations with respect to thickness of shell layers (, , , , ).
Table 9: Variation of natural frequencies (Hz) of a three-layered clamped-clamped cylindrical shell for two configurations with respect to thickness of shell layers (, , , , ).
##### 8.2. Frequency Analysis of Cylindrical Shells on Elastic Foundations

Figure 3 represents variations of natural frequencies (Hz) for three-layered cylindrical shells for SS-SS edge conditions with elastic foundations, that is, ,  N·m. The middle layer is functionally graded material. For both Types 1 and 2 of the cylindrical shells, the frequencies increase considerably by adding the elastic foundations, that is, Pasternak and Winkler models. Also, the values of natural frequencies increase gradually against circumferential wave modes . The minimum frequency is associated with . This shows that the shell vibration is similar to that of beam type for these boundary conditions.

In Table 10, natural frequencies (Hz) for SS-SS three-layered cylindrical shells are listed for Type 1, 2, 3, and 4 cylindrical shells. The shells are based on the Pasternak and Winkler foundations as shown in Figure 2. The values of the shell frequencies show considerable increments by inducting the elastic moduli. Type 1 has the highest frequency, followed by Type 2, Type 4, and Type 3. This also exhibits the effects of material thickness of the layers forming cylindrical shell.

Table 10: Variation of natural frequencies (Hz) of a simply supported-simply supported three-layered cylindrical shell on elastic foundations (, , , , , ,  N·m,  N·m).
Figure 2: Cylindrical shell with elastic foundations.
Figure 3: Variations of natural frequencies (Hz) for three-layered SS-SS cylindrical shells ( N·m, ).

Table 11 displays the variations of natural frequencies (Hz) for four cylindrical shells: Type 1, Type 2, Type 3, and Type 4 for simply supported boundary conditions. The half axial wave mode number (m) is assumed to be one. Here,  N·m and is varied. As is increased, the frequency increases gradually but slowly for the present four types of shells. Also for circumferential wave numbers (n), the frequency increases as n is increased. The minimum frequency corresponds to the circumferential wave number . The vibration becomes beam-type because the fundamental circumferential mode is 1.

Table 11: Variations of natural frequencies (Hz) of SS-SS cylindrical shell for different values of G, where  N·m, , , , .

In Table 12, the natural frequency variations (Hz) for four cylindrical shells Type 1, Type 2, Type 3, and Type 4 are given against circumferential wave numbers (n). The value of Pasternak modulus (G) is maintained and fixed, that is,  N·m for the half axial wave mode . The values of shell frequencies do not increase much with increasing value of . The smallest frequency is associated with , that is, the vibration is similar as that of a beam. Thus the effect of is negligible for keeping constant.

Table 12: Variations of natural frequencies (Hz) of SS-SS cylindrical shell for different values of K, where N·m, , , , .

#### 9. Conclusion

In this study, vibrations of three-layered cylindrical shells are investigated for various shell parameters. Three shell layers are designed by isotropic and functionally graded materials. In the configuration, outer and inner layers are of isotropic layers, whereas the middle layer is assumed to be of functionally graded layers. This forms a combination of isotropic and functionally graded layers. In a shell thickness direction, material composition of a functionally graded material is controlled by volume fraction law. Love shell dynamical equations are considered to describe the vibration problem. Pasternak and Winkler foundations are appended in the transverse direction. The boundary conditions applied on the shell ends are simply supported. Wave propagation approach is utilized to frame the shell frequency equation for a cylindrical shell in the eigenvalue form. The influences of the configurations of the cylindrical shells are analyzed with interchange of materials of shell layers. From this analysis, it is observed that the change of material nature of the shell functionally graded layer influences the shell frequencies. As the thickness of the middle layer is decreased, the vibration frequencies diminish prominently. For cylindrical shells of Types 1 and 2, the vibration frequencies increase considerably by adding the elastic foundations, that is, Pasternak and Winkler models. This analysis can be applied to extract vibration frequencies for the present cylindrical shell configurations for various boundary conditions.

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