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Vibration Analysis of a Three-Layered FGM Cylindrical Shell Including the Effect Of Ring Support

  • Madiha Ghamkhar , Muhammad Nawaz Naeem , Muhammad Imran EMAIL logo and Constantinos Soutis
Published/Copyright: October 6, 2019
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Open Physics
From the journal Open Physics Volume 17 Issue 1

Abstract

In this work, we study vibrations of three-layered cylindrical shells with one ring support along its length. Nature of material of the central layer is a functionally graded material (FGM) type. The considered FGM is of stainless steel and nickel. The internal and external layers are presumed to be made of isotropic material i.e., aluminum. The functionally graded material composition of the center layer is assorted by three volume fraction laws (VFL) which are represented by mathematical expressions of polynomial, exponential and trigonometric functions. The implementation of Rayleigh-Ritz method has been done under the Sanders’ shell theory to obtain the shell frequency equation. Natural frequencies (NFs) are attained for the present model problem under six boundary conditions. Use of characteristic beam functions is made for the estimation of the dependence of axial modals. The impact of layer material variations with ring support is considered for many ring positions. Also the effect of volume fraction laws is investigated upon vibration characteristics. This investigation is performed for various physical parameters. Numerous comparisons of values of shell frequencies have been done with available models of such types of results to verify accuracy of the present formulation and demonstrate its numerical efficiency.

Keywords: cylindrical shell; functionally graded material; isotropic material; ring support; Sanders’ shell theory; Rayleigh-Ritz method
PACS: 02.60.Cb; 68.35.Ja; 02.70.c

1 Introduction

A cylindrical shell is a significant element in structural dynamics. Different mechanical aspects of such types of shells are studied for their practical applications, shell vibration is one of them. This investigation of these shells play a paramount part in the fields of technology, like pressure vessels, nuclear power plants, piping system and other marine and aircraft applications. Many researches [1, 2, 3, 4, 5, 6, 7, 8] have been done the studies on vibrational performance of functionally graded (FG) cylindrical shells (CSs) and influence on the frequencies of layered shells due to edge conditions has been studied by Loy and Lam [9]. The core of all the previous research work has been elaborated by Love’s thin shell theory. Furthermore Loy and Lam [10] gave an investigation of frequency characteristics of thin walled cylindrical shell (CS) under ring supports for several end conditions by using Sanders’ shell theory and Ritz formulation. Xiang et al. [11] used Goldenveizer-Novozhilov theory of shells to determine the exact vibration solution of the same circular cylinders supported by multiple transitional rings. The state-space technique was applied to obtaine shell governing equations for shell splinter and the influences of edge conditions were attained. The effects on frequency parameters for different locations of ring supports were observed. The vibration behavior of open circular cylinders grounded on intermediary ring elastic support was analyzed by Zhang and Xiang [12]. They assessed the influence for number of intermediate ring assistances, their positions, associated boundary conditions and variations included angles on the behavior of the shells. Swaddiwudhipong et al. [13] explained the study of vibration of CSs with middle supports by applying the Ritz method to approximate their frequencies and mode shapes. An analysis of vibration frequency for a FG shell was done by Arshad et al. [14] with effect of different fraction laws by applying Love’s first order shell theory. Another study about the frequency analysis of bi-layered CSs has been presented by the same authors in [15]. The shells were assembled from functionally graded materials (FGMs) as well as isotropic materials. The influences of particular shell configurations on NFs of cylinder-shaped shells were scrutinized. The solidity of the same shells made up of FG structure layer associated with axial load placed on the Winkler-Pasternak foundations was analyzed by Sofiyev and Avear [16]. Arshad et al. [17] explored vibration properties of bi-layered cylinder-shaped shell with both layers made up of FG layers by considering constant thickness. Law-II was exploited to study the material distribution of FGMs. They studied the effect on vibrations of double layered FG shell for various shell constraints, edge conditions and exchange the essential materials making FGMs. Zhang et al. [18] investigated the shell free vibrations for a number of edge conditions by using a differential quadrature type procedure. Naeem et al. [19] analyzed vibration behavior of the tri-layered FGM circular cylinders. They employed the Ritz method and used the Love’s shell theory. Governing mathematical expression was in an integral form by considering the shell strain and kinetic energy relations. Axial modal dependence was examined with solution functions of beam equation. Arshad et al. [20] made a study of FG three-layered cylinder-shaped shells for free vibration under a ring support. Their work deals with the effect of ring supports, located at different positions along the length of cylinder-shaped shell for different edge conditions. The analysis was based on Love’s thin shell theory. Rayleigh-Ritz formulation was employed to obtain solutions of the problem. The vibration response of tri-layered shells was investigated by Li et al. [21]. Ghamkhar et al. [22] studied vibration frequency analysis for three layered cylinder shaped shell with FGM central layer. The effect on shell vibrations for different thickness of the central layer were examined by them. The analysis was based on sander’s shell theory and Ritz mathematical approach. Functionally graded material distribution was controlled with trigonometric volume fraction law.

In this work, vibration frequencies are analyzed for three layered cylindrical shells. These shells are assembled from three layers assuming that the central is made of functionally graded materials, internal and external layers remain isotropic type of materials. Material of the central layer is controlled by following volume fraction laws, polynomial (Law-I), exponential (Law-II) and trigonometric (Law-III). These laws are framed by polynomial, exponential and trigonometric functions. These laws vary the material composition in the radial (through-thickness) direction.

This material variation yields a variety of frequency spectra. Stability of these shells is solidified by ring supports around the tangential directions. Sanders’ thin shell theory is adopted for shell governing equations. These equations are solved by applying Rayleigh Ritz technique involving an energy variation functional. Axial deformation functions are estimated by the solution functions of beam equation. Such functions are taken to meet the edge conditions. An effect of layer thickness configurations is observed on shell natural frequencies. Results are obtained to examine the influence of ring supports at different positions along the shell length.

2 Theoretical considerations

Consider a cylinder-shaped shell sketched in Figure 1. Here length of the shell is denoted byL, thickness is denoted by H and the mean radius symbolized byR. They represent the shell geometrical quantities. A cylindrical coordinate system (x, δ, z) is framed at the shell middle reference surface with x, δ and z as the axial, angular and thickness coordinates respectively. Deformation displacement functions are designated byu1(x, δ, t), u2(x, δ, t) and u3(x, δ, t) which denote the displacement deformations in the longitudinal, tangential and transverse directions respectively. The strain energy ℑ for a thin vibrating CS as in [10] is described below:

Figure 1 Geometry of CS with a ring support that its location varies along the length
Figure 1

Geometry of CS with a ring support that its location varies along the length

(1) ℑ = 1 2 ∫ 0 L ∫ 0 2 π { K } ′ [ C ] { K } R d θ d x

where

(2) { K } ′ = { ∈ 1 , ∈ 2 , ∈ 12 , κ 1 , κ 2 , 2 κ 12 }

and ∈1 , ∈2 and ∈12 denote strains which are related to the reference surface and k1, k2 and k12 represent curvatures. Prime (′) indicates the matrix transposition. The entries of the matrix [C] are furnished as:

(3) [ C ] = x 11 x 12 0 y 11 y 12 0 x 12 x 22 0 y 12 y 22 0 0 0 x 66 0 0 y 66 y 11 y 12 0 z 11 z 12 0 y 12 y 22 0 z 12 z 22 0 0 0 y 66 0 0 z 66

where xij represent the extensional, yij, coupling and zij , bending stiffness. (i, j = 1, 2 and 6). They are defined by the following formulas:

(4) { x i j , y i j , z i j } = ∫ − H / 2 H / 2 Q i j 1 , z , z 2 d z

The reduced stiffness Qij for isotropic materials is stated as [10]:

(5) Q 11 = Q 22 = E ( 1 − μ 2 ) − 1 , Q 12 = μ E ( 1 − μ 2 ) − 1 , Q 66 = E [ 2 ( 1 + μ ) ] − 1

Here E represents the Young’s modulus and μ denotes the Poisson ratio. The matrix yij = 0 for isotropic circular shaped CS and yij ≠ 0 for a FG cylindrical shell; its value determined by the arrangement and properties of its constituent materials. After substituting the expressions from (2) and (3) in (1), ℑ is rewritten as:

(6) ℑ = 1 2 ∫ 0 L ∫ 0 2 π { x 11 ∈ 1 2 + x 22 ∈ 2 2 + 2 x 12 ∈ 1 ∈ 2 + x 66 ∈ 12 2 + 2 y 11 ∈ 1 κ 1 + 2 y 12 ∈ 1 κ 2 + 2 y 12 ∈ 2 κ 1 + 2 y 22 ∈ 2 κ 2 + 4 y 66 ∈ 12 κ 12 + z 11 κ 1 2 + z 22 κ 2 2 + 2 z 12 κ 1 κ 2 + 4 z 66 κ 12 2 } R d θ d x

Following expressions are taken from [23] and written as

(7) { ∈ 1 , ∈ 2 , ∈ 12 } = ∂ u 1 ∂ x , 1 R ∂ u 2 ∂ θ + u 3 , ∂ u 2 ∂ x + 1 R ∂ u 1 ∂ θ
(8) { κ 1 , κ 2 , κ 12 } = − ∂ 2 u 3 ∂ x 2 , − 1 R 2 ∂ 2 u 3 ∂ θ 2 − ∂ u 2 ∂ θ , − 1 R ∂ 2 u 3 ∂ x ∂ θ − 3 4 ∂ u 2 ∂ x + 1 4 R ∂ u 1 ∂ θ .

By substituting expressions (7) and (8) into equation (6), then ℑ becomes as:

(9) ℑ = R 2 ∫ 0 2 π ∫ 0 L x 11 ∂ u 1 ∂ x 2 + x 22 R 2 ∂ u 2 ∂ θ + u 3 2 + 2 x 12 R ∂ u 1 ∂ x ∂ u 2 ∂ θ + u 3 + x 66 ∂ u 2 ∂ x + 1 R ∂ u 1 ∂ θ 2 − 2 y 11 ∂ u 1 ∂ x ∂ 2 u 3 ∂ x 2 − 2 y 12 R 2 ∂ u 1 ∂ x ∂ 2 u 3 ∂ θ 2 − ∂ u 2 ∂ θ − 2 y 12 R ∂ u 2 ∂ θ + u 3 ∂ 2 u 3 ∂ x 2 − 2 y 22 R 3 ∂ u 2 ∂ θ + u 3 ∂ 2 u 3 ∂ θ 2 − ∂ u 2 ∂ θ − 4 y 66 R ∂ u 2 ∂ x + 1 R ∂ u 1 ∂ θ ∂ 2 u 3 ∂ x ∂ θ − 3 4 ∂ u 2 ∂ x + 1 4 R ∂ u 1 ∂ θ + z 11 ∂ 2 u 3 ∂ x 2 2 + z 22 R 4 ∂ 2 u 3 ∂ θ 2 − ∂ u 2 ∂ θ 2 + 2 z 12 R 2 ∂ 2 u 3 ∂ x 2 ∂ 2 u 3 ∂ θ 2 − ∂ u 2 ∂ θ + 4 z 66 R 2 ∂ 2 u 3 ∂ x ∂ θ − 3 4 ∂ u 2 ∂ x + 1 4 R ∂ u 1 ∂ θ 2 d x d θ

The kinetic energy T for a CS is expressed as:

(10) T = 1 2 ∫ 0 L ∫ 0 2 π ρ t ∂ u 1 ∂ t 2 + ∂ u 2 ∂ t 2 + ∂ u 3 ∂ t 2 R d θ d x

here time variable is denoted by t and the mass density per unit length is represented by ρt and is written as:

(11) ρ t = ∫ − H / 2 H / 2 ρ d z

where ρ is mass density.

The Lagrange energy functional Г for a CS is defined as a function of the kinetic and strain energies as:

(12) Γ = T − ℑ

3 Numerical Procedure

The present cylindrical shell is solved by the Rayleigh-Ritz technique. Its deformation displacement fields are expressed in terms of product of functions of space and time variables. These functions for a CS with ring supports can

be assumed in the longitudinal, tangential and transverse directions as:

(13) u 1 ( x , θ , t ) = a m U ( x ) cos ⁡ n θ sin ⁡ ω t u 2 ( x , θ , t ) = b m V ( x ) sin ⁡ n θ sin ⁡ ω t u 3 ( x , θ , t ) = c m W ( x ) cos ⁡ n θ sin ⁡ ω t

here U ( x ) = d ξ ¯ ( x ) d x , V ( x ) = ξ ¯ ( x ) , a n d W ( x ) = ξ ¯ ( x ) ∏ i = 1 P ( x − ai)ℓi and ξ ¯ ( x ) signifies the axial function that fulfills end point conditions. In the longitudinal direction at x = ai, ithring support is present. Here ℓi = 1 and ℓi = 0 represents with and without a ring support respectively. n is the circumferential wave number and ω is the angular vibration frequency. The coefficients am, bm and cm signify the vibration’s amplitudes in the x, θ and z directions respectively and m is the axial half wave number in the axial direction.(x)is chosen for the axial deformation function which is the characteristic beam function as in reference [10]:

(14) ξ ¯ ( x ) = β 1 cosh ⁡ ( α m x ) + β 2 cos ⁡ ( α m x ) − χ m ( β 3 sinh ⁡ ( α m x ) + β 4 sin ⁡ ( α m x ) )

where the values of βi , (i = 1, 2, 3, 4),depend upon the nature of the edge conditions and αm denotes the roots of trigonometric or hyperbolic equations and the parameters, χm ′s depend on values of αm.

Following dimensionless parameters are utilized to simplify the problem.

(15) U _ = U ( x ) H , V _ = V ( x ) H , W _ = W ( x ) R x _ i j = x i j H , y _ i j = y i j H 2 , z _ i j = z i j H 3 α = R / L , β = H / R , X = x L , ρ _ t = ρ t H

Now the expressions (13) are re-written as:

(16) u 1 ( x , θ , t ) = a m H U _ cos ⁡ n θ sin ⁡ ω t u 2 ( x , θ , t ) = b m H V _ sin ⁡ n θ sin ⁡ ω t u 3 ( x , θ , t ) = c m R W _ cos ⁡ n θ sin ⁡ ω t

After making substitutions of the expressions (16) and their respective derivatives into the relations (9) and (10), ℑmax and Tmax are obtained using the principle of conservation of energy. By applying the principle of maximum energy, the Lagrange functional, Гmax takes the following form:

(17) Γ max = π H L R 2 R 2 ω 2 ρ _ t ∫ 0 1 β 2 a m U _ 2 + β 2 b m V _ 2 + c m W _ 2 d X − ∫ 0 1 α 2 β 2 x _ 11 a m d U _ d X 2 + x _ 22 − n β b m V _ + c m W _ 2 + 2 α β x _ 12 a m d U _ d X × − n β b m V _ + c m W _ + x _ 66 α β b m d V _ d X + n β a m U _ 2 − 2 α 3 β 2 y _ 11 a m d U _ d X c m 2 d 2 W _ d X 2 − 2 α β 2 y _ 12 × a m d U _ d X − n 2 c m W _ + n β b m V _ − 2 α 2 β y _ 12 × − n β b m V _ + c m W _ c m 2 d 2 W _ d X 2 − 2 β y _ 22 ( − n β b m V _ + c m W _ ) − n 2 c m W _ + n β b m V _ − 4 β y _ 66 ( α β b m d V _ d X + n β a m U _ ) ⋅ n α c m d W _ d X − 3 α β 4 b m d V _ d X + n β 4 a m U _ + α 4 β 2 z _ 11 c m 2 d 2 W _ d X 2 2 + β 2 z _ 22 − n 2 c m W _ + n β b m V _ 2 + 2 α 2 β 2 z _ 12 c m 2 d 2 W _ d X 2 × − n 2 c m W _ + n β b m V _ + 4 z _ 66 n α c m d W _ d X − 3 α β 4 b m d V _ d X + n β 4 a m U _ 2 d X

4 Formation of eigenvalue frequency equation

The shell eigenvalue frequency equation is derived by making a use of the Rayleigh-Ritz technique. The energy Lagrange functional Гmax is extremized with regard to vibration coefficients: am, bm and cm, we obtain the following relations.

(18) ∂ Γ max ∂ a m = ∂ Γ max ∂ b m = ∂ Γ max ∂ c m = 0

A system of homogeneous simultaneous equations in am, bm and cm is generated and is transformed into the eigenvalue problem as.

(19) K − Ω 2 M X ¯ = 0

where [K] is the stiffness matrix and [M]represents the mass matrix and

(20) Ω 2 = R 2 ω 2 ρ _ t

and

(21) X ¯ ′ = a m , b m , c m

The elements of [K] and [M] are given in the Appendix 1. MATLAB software is used to solve the eigenvalue problem (19) for the shell frequency spectra for various physical parameters.

5 Functionally graded materials

In practice of three layered cylindrical shell, its central layer is fabricated by FGMs and isotropic is used for internal and external layers as shown in Figure 2. Here the stiffness moduli are altered as:

Figure 2 Cross-section of three-layered CS.
Figure 2

Cross-section of three-layered CS.

(22) x i j = x i j i n t ( I ) + x i j c e n ( F ) + x i j e x t ( I ) y i j = y i j i n t ( I ) + y i j c e n ( F ) + y i j e x t ( I ) z i j = z i j i n t ( I ) + z i j c e n ( F ) + z i j e x t ( I )

where i,j=1, 2, 6 and superscript int(I), ext(I) represent the isotropic internal and external layers and cen(F) denotes the central FGM layer. The functionally graded materials contain two essential materials. These materials are stainless steel and nickel. The material parameters for stainless steel material are: E2, μ2,ρ2 and for nickel material are: E1, μ1,ρ1. The thickness of each layer is presumed to be H/3. Then the actual material quantities for FGM layer are

given as:

(23a) E F = E 1 − E 2 6 z + H 2 H N + E 2
(23b) μ F = μ 1 − μ 2 6 z + H 2 H N + μ 2
(23c) ρ F = ρ 1 − ρ 2 6 z + H 2 H N + ρ 2

The material properties for middle FGM layer vary from z = −H/6 to H/6. From the relations (23a-c), the effective material properties become EF = E2, μF = μ2 and ρF = ρ2 at z = −H/6 where for z = H/6 material properties becomeEF = E1, μF = μ1 and ρF = ρ1. Thus forz = −H/6 , the shell is contained only stainless steel whereas for z = H/6 consisted of nickel material. In a FGM shell, the distribution of materials is controlled by various volume fraction laws. Three volume fraction laws are expressed in mathematical form. If z symbolizes the basic shell thickness variable then the volume fraction law VF of a FGM is formulated as following function [24]

(24) V F = 6 z + H 2 H N

where H represents the thickness of cylinder-shaped shell and N denotes the power law proponent which may take values from zero to infinity. A volume fraction law formulated by Arshad et al. [14] as:

(25) V F = 1 − e − 6 z + H 2 H N

where e be the standard irrational natural exponential base number. The material properties are written as:

(25a) E F = E 1 − E 2 1 − e − 6 z + H 2 H N + E 2
(25b) μ F = μ 1 − μ 2 1 − e − 6 z + H 2 H N + μ 2
(25c) ρ F = ρ 1 − ρ 2 1 − e − 6 z + H 2 H N + ρ 2

Trigonometric volume fraction law for a FGM circular CS is stated as:

(26) V F 1 = sin 2 6 z + H 2 H N , V F 2 = cos 2 6 z + H 2 H N

Here

(26a) V F 1 + V F 2 = 1

The material parameters for FG cylindrical shell are written as:

(26b) E F = E 1 − E 2 sin 2 6 z + H 2 H N + E 2
(26c) μ F = μ 1 − μ 2 sin 2 6 z + H 2 H N + μ 2
(26d) ρ F = ρ 1 − ρ 2 sin 2 6 z + H 2 H N + ρ 2

6 Results and discussion

To check the validity of the current work, results for simply supported and clamped CSs with no ring support are compared with others available in the literature. A good agreement is found among the present results and those obtained by other techniques. In Table 1, a comparison of frequency parameters Δ = ω R 1 − μ 2 ρ / E for simply supported isotropic CS is presented with those in Zhang et al. [18]. Table 3 represents the NFs (Hz) for a simply supported FGM cylindrical shell of type 1 compared with those obtained by Loy et al. [1] and for power law exponents N = 0.5, 1, 2. The frequency at n = 3 which is about 0.9%, 0.5%, and 0.1% minimum than those available in [1]. It is determined from thecomparison of NFs that the current method is efficient and gives accurate results. Two types of CS are described in Table 4.where MA, MB and MC represent Aluminum, Stainless Steel and Nickel respectively. Material properties of the isotropic material as well as FGM constituents are given in reference [1] and [2]. Different layer thicknesses were used for the analysis of three-layered FGM cylindrical shell as shown in Table 5.

Table 1

Comparison of frequency parameter Δ = ω R 1 − μ 2 ρ / E for simply-supported isotropic CS. (μ = 0.3, m = 1, H/R = 0.05)

L/R n n n n
1 2 3 4
20 Zhang et al.[18] 0.016102 0.039271 0.109811 0.210277
20 Present 0.0161029 0.0392713 0.1098115 0.2102771
20 Difference % 0.006 0.001 0.001 0.000
Table 2

Comparison of frequency parameter Δ = ω R 1 − μ 2 ρ / E for a clamped isotropic shell (μ = 0.3, H/R = 0.05)

L/R n n n n
1 2 3 4
20 Zhang et al.[18] 0.03285 0.040638 0.109973 0.210324
20 Present 0.03440 0.040772 0.110005 0.210376
20 Difference % 4.7 0.33 0.03 0.02
Table 3

Comparison of natural frequencies (Hz) for type I FGM cylindrical shell with simply supported edges. (m = 1, H/R = 0.002, L/R = 20)

Loy et al. [1] Present
N N
n 0.5 1 2 0.5 1 2
1 13.321 13.211 13.103 13.321 13.210 13.103
2 4.5168 4.4800 4.4435 4.5098 4.4746 4.4396
3 4.1911 4.1569 4.1235 4.1520 4.1356 4.1154
4 7.0972 7.0384 6.9820 7.0189 7.0000 6.9721
5 11.336 11.241 11.151 11.210 11.181 11.138
Table 4

Types of shell w. r. t the arrangements of shell layers

Types Internal Layer Central FGM Layer External Layer
Type I MA MB/MC MA
Type II MA MC/MB MA
Table 5

Thickness variation of the FGM shell layers

Thickness Patterns Inner isotropic Layer Central FGM Layer External isotropic Layer
Case 1 H/3 H/3 H/3
Case 2 H/4 H/2 H/4
Case 3 H/5 3H/5 H/5

Table 6-8 show the variation of NFs (Hz) versus n for three layered FGMs type I CS with ring supports. Thickness of the center layer is presumed to be H/3, H/2 and 3H/5 for Tables 6-8 respectively. In these Tables, the influence of three VFL is perceived for six edge conditions: simply supported-simply supported (SS − SS), clamped-clamped (C−C), free-free (F−F), clamped-simply supported (C−SS), clamped-free (C − F) and free-simply supported (F − SS).It is noticed that the NF increased with the increase of n. It is also examined that the C − C edge condition has the maximum NFs (Hz) and C − F has the really minimum. It is studied that natural frequencies increasing swiftly from n is equal to 1 to 2 then its increasing steadily. In Table 6, Law-I gets the extreme frequencies (Hz) and Law-III takes the lowest frequencies (Hz). Table 7 & 8 represent the variation of NFs (Hz) of FGM shell by using Law-I for six boundary conditions.

Table 6

Variation of NFs (Hz) for Type I & Case 1 FGM CS against n with ring support. (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1)

n SS − SS C − C F − F C − SS C − F F − SS
Law I 1 505.393 513.611 503.363 507.878 278.876 503.485
2 848.181 848.183 48.182 762.667 349.123 817.553
3 848.297 848.3 48.299 773.636 367.637 822.354
4 848.604 848.61 848.608 777.258 375.168 823.97
5 849.247 849.256 849.253 779.396 379.799 825.189
6 850.411 850.424 850.42 781.42 384.072 826.677
7 852.321 852.338 852.333 783.941 389.226 828.811
8 855.239 855.261 855.254 787.392 396.115 831.918
9 859.46 859.488 859.479 792.16 405.447 836.328
10 865.311 865.346 865.335 798.631 417.851 842.389
Law II 1 504.242 512.441 502.216 506.72 278.238 502.338
2 846.248 846.25 846.249 760.929 348.327 815.69
3 846.377 846.38 846.379 771.884 366.804 820.492
4 846.701 846.706 846.705 775.514 374.326 822.122
5 847.366 847.374 847.372 777.669 378.96 823.36
6 848.556 848.569 848.565 779.715 383.24 824.874
7 850.497 850.514 850.509 782.264 388.404 827.038
8 853.45 853.472 853.465 785.747 395.305 830.179
9 857.711 857.739 857.73 790.55 404.65 834.627
10 863.605 863.64 863.629 797.06 417.066 840.73
Law III 1 503.416 511.602 501.394 505.891 277.779 501.515
2 844.84 844.841 844.841 759.663 347.748 814.332
3 844.949 844.952 844.951 770.583 366.186 819.108
4 845.246 845.252 845.25 774.182 373.684 820.709
5 845.875 845.884 845.881 776.302 378.294 821.913
6 847.022 847.035 847.031 778.307 382.549 823.383
7 848.911 848.928 848.922 780.806 387.683 825.495
8 851.802 851.825 851.817 784.231 394.548 828.577
9 855.993 856.021 856.012 788.969 403.85 832.956
10 861.808 861.843 861.831 795.404 416.218 838.981
Table 7

Variation of NFs (Hz) for Type I & Case 2 FGM CS against n with ring support. (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1)

n SS − SS C − C F − F C − SS C − F F − SS
1 499.678 507.874 497.709 502.171 276.964 497.805
2 842.358 842.36 842.359 757.432 346.726 811.926
3 842.477 842.481 842.48 768.329 365.114 816.711
4 842.788 842.793 842.792 771.93 372.594 818.322
5 843.433 843.441 843.438 774.059 377.193 819.538
6 844.595 844.608 844.603 776.074 381.434 821.022
7 846.498 846.515 846.509 778.583 386.546 823.147
8 849.4 849.422 849.415 782.012 393.376 826.237
9 853.595 853.623 853.613 786.748 402.625 830.619
10 859.406 859.44 859.428 793.172 414.918 836.637
Table 8

Variation of NFs (Hz) for Type I & Case 3 FGM CS against n with ring support. (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1)

n SS − SS C − C F − F C − SS C − F F − SS
1 496.300 504.482 494.366 498.797 275.838 494.447
2 838.929 838.931 838.930 754.349 345.315 808.612
3 839.052 839.055 839.054 765.205 363.630 813.389
4 839.367 839.372 839.371 768.797 371.083 815.000
5 840.017 840.026 840.023 770.925 375.671 816.220
6 841.186 841.199 841.195 772.943 379.906 817.709
7 843.096 843.113 843.108 775.456 385.014 819.840
8 846.006 846.028 846.021 778.891 391.839 822.937
9 850.209 850.236 850.227 783.633 401.082 827.326
10 856.027 856.061 856.050 790.062 413.365 833.350

Table 9-11 describe the variation of NFs (Hz), effects of shell configurations on NFs (Hz) versus n for FGM shell type II with ring supports. Thickness of the center layer is supposed as same as for Table 6, 7 and 8 respectively. Law-I gets the lowest values and Law-III takes extreme values of NFs (Hz) in Table 9. Table 12 represents the natural frequency (Hz) of three layered CS of case1 versus n with and without ring support for SS−SS boundary conditions. The behavior of frequency remains same according to the circumferential wave number n and volume fraction laws for both CSs with and without ring support. Table 12 shows that the NFs of CS with ring support are higher than frequencies of the CS without ring support.

Table 9

Variation of NFs (Hz) for Type II & Case 1 FGM CS against n with ring support. (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1)

n SS − SS C − C F − F C − SS C − F F − SS
Law I 1 505.880 514.077 503.833 508.352 278.623 503.965
2 847.410 847.411 847.411 761.974 348.805 816.815
3 847.519 847.522 847.521 772.927 367.301 821.601
4 847.817 847.823 847.821 776.537 374.822 823.206
5 848.448 848.457 848.455 778.664 379.448 824.413
6 849.598 849.612 849.608 780.676 383.719 825.889
7 851.492 851.513 851.508 783.185 388.875 828.010
8 854.393 854.422 854.415 786.624 395.770 831.104
9 858.598 858.636 858.628 791.383 405.114 835.503
10 864.433 864.483 864.473 797.847 417.537 841.553
Law II 1 507.040 515.256 504.988 509.517 279.264 505.120
2 849.352 849.353 849.353 763.720 349.605 818.687
3 849.448 849.452 849.451 774.687 368.137 823.471
4 849.729 849.735 849.733 778.289 375.667 825.063
5 850.339 850.348 850.345 780.399 380.291 826.250
6 851.463 851.477 851.472 782.388 384.555 827.701
7 853.329 853.346 853.341 784.870 389.701 829.792
8 856.197 856.220 856.213 788.277 396.584 832.852
9 860.367 860.395 860.386 793.000 405.916 837.212
10 866.165 866.199 866.188 799.426 418.327 843.222
Law III 1 507.880 516.109 505.824 510.361 279.730 505.956
2 850.780 850.782 850.781 765.004 350.192 820.064
3 850.896 850.899 850.898 776.006 368.763 824.874
4 851.204 851.210 851.208 779.639 376.318 826.495
5 851.849 851.858 851.855 781.785 380.965 827.717
6 853.018 853.030 853.026 783.815 385.255 829.211
7 854.935 854.953 854.947 786.347 390.431 831.354
8 857.865 857.887 857.880 789.812 397.350 834.474
9 862.105 862.133 862.124 794.601 406.725 838.903
10 867.982 868.017 868.006 801.101 419.185 844.991
Table 10

Variation of NFs (Hz) for Type II & Case 2 FGM CS against n with ring support. (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1).

n SS − SS C − C F − F C − SS C − F F − SS
1 500.409 508.573 498.414 502.883 276.586 498.525
2 841.206 841.208 841.207 756.396 346.253 810.824
3 841.310 841.313 841.312 767.265 364.611 815.580
4 841.600 841.606 841.604 770.843 372.076 817.169
5 842.221 842.230 842.227 772.949 376.667 818.362
6 843.356 843.369 843.365 774.940 380.907 819.820
7 845.231 845.248 845.243 777.424 386.029 821.918
8 848.107 848.129 848.122 780.833 392.880 824.983
9 852.279 852.307 852.298 785.553 402.168 829.344
10 858.073 858.108 858.096 791.967 414.518 835.348
Table 11

Variation of NFs (Hz) for Type II & Case 3 FGM CS against n with ring support.(L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1).

n SS − SS C − C F − F C − SS C − F F − SS
1 497.177 505.321 495.213 499.652 275.387 495.311
2 837.550 837.551 837.551 753.109 344.748 807.294
3 837.650 837.653 837.652 763.926 363.024 812.031
4 837.933 837.938 837.936 767.484 370.452 813.608
5 838.542 838.551 838.548 769.572 375.016 814.787
6 839.661 839.674 839.670 771.543 379.227 816.227
7 841.513 841.530 841.524 774.002 384.310 818.302
8 844.356 844.378 844.371 777.376 391.108 821.334
9 848.485 848.513 848.504 782.050 400.323 825.651
10 854.223 854.257 854.246 788.406 412.578 831.597
Table 12

Variation of NFs (Hz) of three layered FGM CS Type I & Case1 versus circumferential wave number (n). (L/R = 50, m = 1, H/R = 0.007, a = 0.5, N = 1).

without ring support with ring support
n Type I Type I
1 1.6289 505.393
2 4.6222 848.181
3 12.991 848.297
4 24.903 848.604
5 40.271 849.247
6 59.076 850.411
7 81.309 852.321
8 106.968 855.239
9 136.052 859.460
10 168.559 865.311

Table 13 demonstrates NFs (Hz) with ratios (L/R) for type I & case 1 FGM shell. The natural frequencies decreased less than 0.5%with the increasing values of N.Natural frequencies decreased 11% and 14% when L/R becomes 10 and 20 respectively.

Table 13

Natural Frequencies (Hz) of Type I & Case 1 FGM CS with ring support versus length to radius ratios L/R. (n = 1, m = 1, , H/R = 0.005, a = 0.5).

L/R N=1 N=2 N=3 N=4 N=5 N=10 N=20 N=30 N=50
5 595.701 593.988 593.135 592.625 592.285 591.513 591.067 590.905 590.763
10 528.705 527.184 526.427 525.973 525.672 524.985 524.587 524.440 524.309
20 510.584 509.115 508.384 507.946 507.655 506.991 506.607 506.464 506.337

The thickness of each layer of the shell for Figure 3-8 is H/3. Natural frequency varies with respect to the ring support position and this influence changes according to edge conditions. Figure 3 shows the variation of NF of SS − SS shell with the position of ‘a’ for different L/R ratios. Natural frequencies are obtained for law-I, law-II and law-III. The movements of these VFL, for frequency curve at a = 0, the values are 328.96 , 328.51, 327.88, at a = 0.5, the values are 593.99, 593.17, 592.85 and at a = 1,values are 328.97, 328.51, 327.89 for law-I, law-II and law-III respectively. The behavior of the frequency curve is increasing from a = 0 to a = 0.5 and decreasing from a = 0.5 to a = 1. So it is a symmetric curve. Similar behavior studied for L/R =10, 20 and also for C − C and F − F edge conditions. Moreover, law-II is consisted between the law-I & III. Frequency curves are overlapping because the values for all laws are so closed to each other. So for other boundary conditions law-III is selected to draw because it attains minimum frequency values. Figure 4 shows the same results for clamped-clamped edge condition. Frequencies are significantly high for clamped-clamped end condition. In Figure 5 variation of NFs (Hz) of three-layered FGM cylindrical shell is plotted against the ring supports position. So the frequency value at a = 0, is 260.46 and the extreme NF (Hz) for law-III lies at a = 0.6, frequency is 683.25. The last value of frequency curve is 278.24 lies at a = 1,. Similar behavior displays for L/R = 10 and 20. In Figure 6. The frequency curve is increasing gradually from a = 0 to 0.5 then gets its extreme value at a = 0.8, after this curve starts to decrease. Here the frequency curve is not chime formed because of different edge condition. It is noticed that the behavior of NFs curves for all ratios and laws are same. Figures 7 and 8 exhibits variation of NFs (Hz) versus n for different N with a = 0.5 for SS−SS and C−C end point conditions. Here N differs as 0.5, 1 and 2. Here frequency curves increase rapidly from n = 1 to 2 and then these curves start to increase linearly through n. It is noticed that with the increase in power law exponent N natural frequency is not really affected.

Figure 3 Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L/R ratios for SS − SS edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)
Figure 3

Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L/R ratios for SS − SS edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)

Figure 4 Variation of NFs (Hz) of three-layered FGM CS with ring supports position ‘a’ at different L /R ratios for Law-I, C − C edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)
Figure 4

Variation of NFs (Hz) of three-layered FGM CS with ring supports position ‘a’ at different L /R ratios for Law-I, C − C edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)

Figure 5 Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L /R ratios for C−SS edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)
Figure 5

Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L /R ratios for C−SS edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)

Figure 6 Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L /R ratios for C − F edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)
Figure 6

Variation of NFs (Hz) of three-layered FGM CS versus ring supports position ‘a’ at different L /R ratios for C − F edge conditions. (m = 1, n = 1, H/R = 0.005 and N = 2)

Figure 7 Variation of NFs (Hz) of FGM cylindrical shell versus (n) for different N with SS − SS edge conditions. (m=1, L/R=50, H/R=0.007)
Figure 7

Variation of NFs (Hz) of FGM cylindrical shell versus (n) for different N with SS − SS edge conditions. (m=1, L/R=50, H/R=0.007)

Figure 8 Variation of NFs (Hz) of FGM cylindrical shell versus (n) for different N with C − C edge conditions. (m=1, L/R=50, H/R=0.007)
Figure 8

Variation of NFs (Hz) of FGM cylindrical shell versus (n) for different N with C − C edge conditions. (m=1, L/R=50, H/R=0.007)

7 Conclusions

The frequency analysis of three-layered FGM cylindrical shell is performed to determine the effect of ring support. The shell central layer is made of FGMs while the internal and external layers are of isotropic material. Variation of NFs (Hz) is analyzed for six boundary conditions. It is concluded that the material dissemination controlled by the VFL which has little effect (<1%) on vibration frequency of a FGM CS but law-III is recommended for type 1 FGM shell and law-I is for type 2 FGM shell to estimate the lower frequency values.

Natural frequencies are increased with the increase of n and decreased with the increase of L/R ratios. Natural frequency also decreased <2% when increase in the thickness of the central layer becomes double. The frequency curve of the shell with ring support at different positions get symmetric shapes because of same edge conditions. They are not symmetrical about center because of different end point conditions. The induction of ring support on cylinder-shaped shell has significant effect on the NFs as compared to the shell frequencies without ring support.

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Appendix

K 11 = α 2 β 2 x 11 I 1 + n 2 β 2 ( x 66 − β y 66 + β 2 z 66 / 4 ) I 2 K 12 = − n α β 2 ( x 12 + β y 12 ) I 3 + n α β 2 ( x 66 + β y 66 − 3 β 2 z 66 / 4 ) I 4 K 13 = α β ( x 12 + n 2 β y 12 ) I 5 + n 2 α β 2 ( − 2 y 66 + β z 66 ) I 6 − α 3 β 2 y 11 I 7 K 22 = n 2 β 2 ( x 22 + 2 β y 22 + β 2 z 22 ) I 8 + α 2 β 2 ( x 66 + 3 β y 66 + 9 β 2 z 66 / 4 ) I 9 K 21 = K 12 K 23 = − n β ( x 22 + ( n 2 + 1 ) β y 22 + n 2 β 2 z 22 ) I 10 − n α 2 β 2 ( 2 y 66 + 3 β z 66 ) I 11 + n α 2 β 2 ( y 12 + β z 12 ) I 12 K 31 = K 13 K 32 = K 23 K 33 = ( x 22 + 2 n 2 β y 22 + n 4 β 2 z 22 ) I 13 + 4 n 2 α 2 β 2 z 66 I 14 − α 2 β ( y 12 + n 2 β z 12 ) I 15 + α 4 β 2 z 11 I 16

where

I 1 = ∫ 0 L d U _ / d X 2 d X , I 2 = ∫ 0 L U _ 2 d X , I 3 = ∫ 0 L d U _ / d X V _ d X , I 4 = ∫ 0 L d V _ / d X U _ d X , I 5 = ∫ 0 L d U _ / d X W _ d X , I 6 = ∫ 0 L U _ d W _ / d X d X , I 7 = ∫ 0 L d U _ / d X d 2 W _ / d X 2 d X , I 8 = ∫ 0 L V _ 2 d X , I 9 = ∫ 0 L d V _ / d X 2 U _ d X , I 10 = ∫ 0 L V W _ d X , I 11 = ∫ 0 L d V _ / d X d W _ / d X d X , I 12 = ∫ 0 L V _ d 2 W _ / d X 2 d X , I 13 = ∫ 0 L W 2 _ d X , I 14 = ∫ 0 L d W _ / d X 2 d X I 15 = ∫ 0 L 2 d 2 W _ / d X 2 W _ d X ,
I 16 = ∫ 0 L ( d 2 W _ / d X 2 ) 2 dX,
x 11 = 2 E 3 ( 1 − μ 2 ) + 1 ( 1 − μ 1 2 ) E 1 − E 2 3 N + 1 + E 2 3 x 12 = 2 E μ 3 ( 1 − μ 2 ) + μ 1 ( 1 − μ 1 2 ) E 1 − E 2 3 N + 1 + E 2 3 x 66 = E 3 ( 1 + μ ) + 1 2 ( 1 + μ 1 ) E 1 − E 2 3 N + 1 + E 2 3 x 22 = x 11
x 11 = 2 E 3 ( 1 − μ 2 ) + 1 ( 1 − μ 1 2 ) E 1 − E 2 3 N + 1 + E 2 3 x 12 = 2 E μ 3 ( 1 − μ 2 ) + μ 1 ( 1 − μ 1 2 ) E 1 − E 2 3 N + 1 + E 2 3 x 66 = E 3 ( 1 + μ ) + 1 2 ( 1 + μ 1 ) E 1 − E 2 3 N + 1 + E 2 3 x 22 = x 11
y 11 = E 1 − E 2 ( 1 − μ 1 2 ) 1 18 N + 1 + 1 9 N + 1 N + 2 y 12 = μ 1 ( E 1 − E 2 ) ( 1 − μ 1 2 ) 1 18 N + 1 + 1 9 N + 1 N + 2 y 66 = ( E 1 − E 2 ) 2 ( 1 + μ 1 ) 1 18 N + 1 + 1 9 N + 1 N + 2 y 22 = y 11
z 11 = z 22 = 13 E 162 ( 1 − μ 2 ) + E 1 − E 2 ( 1 − μ 1 2 ) × 1 108 N + 1 − 1 27 N + 1 N + 2 − 2 27 N + 1 N + 2 ( N + 3 ) + E 2 324 ( 1 − μ 1 2 ) z 12 = 13 μ E 162 ( 1 − μ 2 ) + μ 1 ( E 1 − E 2 ) ( 1 − μ 1 2 ) × 1 108 N + 1 − 1 27 N + 1 N + 2 − 2 27 N + 1 N + 2 ( N + 3 ) + μ 1 E 2 324 ( 1 − μ 1 2 )
z 66 = 13 E 324 ( 1 + μ ) + E 1 − E 2 2 ( 1 + μ 1 ) × 1 108 N + 1 − 1 27 N + 1 N + 2 − 2 27 N + 1 N + 2 ( N + 3 ) + E 2 648 ( 1 + μ 1 )
M 11 = β 2 I 2 , M 22 = β 2 I 8 , M 33 = I 13 M 12 = M 13 = M 21 = M 23 = M 31 = M 32 = 0
Received: 2018-06-28
Accepted: 2019-07-27
Published Online: 2019-10-06

© 2019 M. Ghamkhar et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  112. Erratum to: Energy converting layers for thin-film flexible photovoltaic structures
https://doi.org/10.1515/phys-2019-0060
Keywords for this article
cylindrical shell; functionally graded material; isotropic material; ring support; Sanders’ shell theory; Rayleigh-Ritz method
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