American Journal of Applied Mathematics
Volume 3, Issue 3-1, June 2015, Pages: 32-40

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

Email address:

(Z. G. Bhutta)
(M. N. Naeem)
(M. Imran)

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


1. Introduction

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. [1] examined free vibrations of orthogonally inflexible cylindrical shells where rigidness has been treated as distinct elements. Sharma et al. [2] 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. [3]. Vibrations of functionally graded (FG) cylindrical shells were investigated by Loy et al.[4] and Pardhan et al. [5] for various physical parameters and several boundary conditions. Li et al. [6] 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 [7] for studying axial buckling of cylindrical shells and they investigated this aspect of dynamical study of the shells. Bing et al. [8] examined vibration frequencies of thin walled cylindrical shells for different edge condition. Shao and Ma [9] 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. [10] employed the Ritz formulation to investigate vibration of natural frequency characteristic of FG cylindrical shells. Naeem et al. [11] 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. [12] in which one layer was FG layer and other layer was of homogeneous materials. Iqbal et al. [13] 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. [14]. Sofiyev et al. [15] examined the non-linear free vibration of FG cylindrical shells attached to combine loads with various ends conditions and resting on elastic foundations. Vel [16] 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. [17] applied exponential volume fraction law to observe the cylindrical shell’s vibration with FGM. Warburton et al. [18] 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 [19]. 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. [20]. 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.[21] and Koizumi [22] 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.

 

Figure 1. Coordinate system and shell geometry.

For a vibrating thin cylindrical shell, its strain energy, expressed by U is stated as Loy et al. [4]:

 

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 [4]:

    

The reduced material stiffness andfor isotropic materials are described as [4]::

  

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 [4]

   

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:

       

where

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

(11)

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

i.e.              

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. [10] 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:

       

where and

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. [4] is composed in Table 1. The present case was solved by the Raleigh-Ritz method while the frequency parameters in Loy et.al. [4] 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. [4] Present
1 0.016101 0.016101
2 0.009382 0.009363
3 0.022105 0.022085
4 0.042095 0.042075
5 0.068008 0.069788

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).

n N Warburton[18] Present
2 1 2946.8 2042.7
2 5637.8 5631.9
3 8935.3 8926.4
4 11405 1139.3
5 13245 13243.7
3 1 2199.3 2194.4
2 4041.9 4031.2
3 6620.0 6605.9
4 9124.0 9108.4
5 11357 11343.4

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 [18] 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. [13] 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. [13] 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.

Table 3. Natural frequencies (Hz) comparisons of 1st Type cylindrical shells having simply supported – simply supported end condition (m=1, L/R=20, h/R=0.002).

  Iqbal et al. [13] Present
n N=0.5 N=1 N=15 N=0.5 N=1 N=15
1 13.103 13.211 13.505 13.102 13.209 13.504
2 4.4382 4.4742 4.5759 4.4386 4.4742 4.5767
3 4.1152 4.1486 4.2451 4.1256 4.1578 4.2522
4 6.9754 7.0330 7.1943 6.9945 7.0497 7.2051
5 11.145 11.238 11.494 11.172 11.2611 11.5070

Table 4. Natural frequencies (Hz) comparisons of 2nd Type cylindrical shells having simply supported-simply supported end condition. (m=1, L/R=20, h/R=0.002).

  Iqbal et al. [13] Present
n N=0.5 N=1 N=15 N=0.5 N=1 N=15
1 13.321 13.211 12.933 13.321 13.211 12.932
2 4.5168 4.480 4.3834 4.5195 4.4831 4.3858
3 4.1911 4.1569 4.0653 4.2014 4.1685 4.0788
4 7.0972 7.0384 6.8856 7.113 7.0563 6.9091
5 11.336 11.241 10.999 11.356 11.265 11.032

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.

Table 5. Variation of natural frequencies (Hz) of 1st Type with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1) with polynomial fraction law.

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 13.574 22.325 13.511 22.150 13.377 21.678
2 4.4803 7.5297 4.4653 7.4698 4.4052 7.3080
3 3.4889 4.1355 3.4437 4.0902 3.3525 3.9671
4 5.5134 4.4917 5.4206 4.4126 5.2341 4.1950
5 8.7339 6.5838 8.5821 6.4538 8.2779 6.0945
6 12.768 9.4979 12.545 9.3066 12.098 8.7771
7 17.558 13.023 17.251 12.759 16.636 12.029
8 23.092 17.113 22.687 16.766 21.878 15.806
9 29.366 21.756 28.851 21.315 27.822 20.093
10 36.379 26.948 35.741 26.401 34.466 24.888

Table 6. Variation of natural frequencies (Hz) of 1st Type with C-F and C-SS boundary condition (L=20, h=0.002, R=1, m=1) with polynomial fraction law.

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 22.325 10.057 22.150 9.9785 21.811 9.8255
2 7.5302 3.3181 7.4704 3.2903 7.3540 3.2356
3 4.1365 2.5839 4.0915 2.5433 4.0028 2.4624
4 4.4927 4.0833 4.4139 4.0032 4.2580 3.8445
5 6.5845 6.4683 6.4548 6.3380 6.1980 6.0802
6 9.4984 9.4564 9.3072 9.2650 8.9292 8.8865
7 13.033 13.004 12.759 12.740 12.239 12.219
8 17.113 17.102 16.766 16.755 16.081 16.070
9 21.756 21.748 21.315 21.307 20.443 20.435
10 26.948 26.942 26.402 26.396 25.322 25.316

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

Table 7. Variation of natural frequencies (Hz) of 2nd Type with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1) with polynomial fraction law.

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 13.442 21.950 13.473 13.473 13.647 22.474
2 4.4217 7.3967 4.4318 4.4318 4.4890 7.5161
3 3.3347 3.9608 3.3418 3.3418 3.3823 4.0957
4 5.1848 4.0480 5.1956 5.1956 5.2558 4.2878
5 8.1969 5.8092 8.2140 8.2140 8.3084 6.2076
6 11.980 8.3458 12.005 12.005 12.143 8.9338
7 16.474 11.432 16.508 16.508 16.697 12.242
8 21.665 15.018 21.710 21.710 21.959 16.084
9 27.551 19.090 27.609 27.609 27.925 20.447
10 34.130 23.645 34.202 34.202 34.593 25.327

Table 8. Variation of natural frequencies (Hz) of 2nd Type with C-F and C-SS boundary condition (L=20, h=0.002, R=1, m=1) with polynomial fraction law.

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 21.950 9.8997 22.121 9.9768 22.474 10.136
2 7.3962 3.2569 7.4546 3.2819 7.5757 3.3345
3 3.9597 2.4573 4.0036 2.4753 4.0947 2.5136
4 4.0469 3.8202 4.1258 3.8478 4.2868 3.9054
5 6.1575 6.0386 6.2023 6.0823 6.2946 6.1728
6 8.8670 8.8249 8.9313 8.8888 9.0639 9.0208
7 12.153 12.134 12.240 12.222 12.422 12.403
8 15.968 15.957 18.083 16.073 16.321 16.311
9 20.299 20.292 20.446 20.439 20.749 20.742
10 25.148 25.138 25.325 25.320 25.700 25.695

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.

Table 9. Variation of natural frequencies (Hz) of 1st Type with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 13.374 22.317 13.311 22.149 13.186 21.821
2 3.3600 7.5349 3.3438 7.4782 3.3104 7.3677
3 1.2446 4.0913 1.2388 4.0608 1.2245 4.0014
4 4.1349 4.3080 4.1166 4.2765 4.0800 4.2154
5 7.6059 6.2464 7.5722 6.2010 7.5057 6.1132
6 11.634 8.9922 11.582 8.9269 11.480 8.8007
7 16.288 12.323 16.216 12.234 16.074 12.061
8 21.602 16.192 21.505 16.075 21.317 15.848
9 27.591 20.585 27.468 20.436 27.228 20.147
10 34.264 25.498 34.112 25.313 33.814 24.956

Table 10. Variation of natural frequencies (Hz) of 1st Type with C-F and C-SS boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 22.235 9.9054 22.067 9.8309 21.740 9.6853
2 7.1317 2.4884 7.0792 2.4695 6.9698 2.4315
3 3.1758 0.9217 3.1552 0.9149 3.0966 0.8994
4 3.4062 3.0623 3.3847 3.0402 3.3247 2.9968
5 5.6490 5.6330 5.6102 5.5922 5.5236 5.5130
6 8.5831 8.6162 8.5223 8.5538 8.3971 8.4328
7 12.026 12.063 11.940 11.975 11.768 11.806
8 15.966 15.998 15.851 15.882 15.624 15.658
9 20.407 20.434 20.260 20.286 19.972 19.999
10 25.355 25.376 25.171 25.192 24.814 24.836

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.

Table 11. Variation of natural frequencies (Hz) of 2ndType with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 13.248 21.959 13.311 22.123 13.439 22.465
2 3.3257 7.4147 3.3418 7.4702 3.3760 7.5853
3 1.2283 4.0281 1.2340 4.0580 1.2487 4.1199
4 4.0992 4.2461 4.1172 4.2770 4.1548 4.3407
5 7.5417 6.1591 7.5751 6.2035 7.6433 6.2952
6 11.536 8.8671 11.587 8.9310 11.691 9.0628
7 16.151 12.152 16.223 12.240 16.369 12.420
8 21.421 15.267 21.516 16.083 21.709 16.320
9 27.360 20.299 27.481 20.446 27.728 20.747
10 33.978 25.144 34.128 25.326 34.434 25.699

Table 12. Variation of natural frequencies (Hz) of 2ndType with C-F and CC-SS boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 21.8769 9.7576 22.0404 9.8306 22.3811 9.9825
2 7.0099 2.4494 7.0614 2.4680 7.1716 2.5075
3 3.1066 0.9046 3.1269 0.9113 3.1777 0.9274
4 3.3397 3.0190 3.3610 3.0406 3.4134 3.0860
5 5.5596 5.5544 5.5978 5.5943 5.6818 5.6771
6 8.4568 8.4964 8.5165 8.5575 8.6430 8.6839
7 11.8545 11.8957 11.9390 11.9813 12.1157 12.1581
8 15.7410 15.7764 15.8537 15.8901 16.0880 16.1245
9 20.1214 20.1506 20.2658 20.2957 20.5650 20.5951
10 25.0006 25.0244 25.1803 25.2047 25.5519 25.5764

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.

Table 13. Variation of natural frequencies (Hz) of 1st Type with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 12.7887 21.1528 12.7325 20.9910 12.6077 20.6817
2 4.2153 7.0933 4.1950 7.0391 4.1552 6.9355
3 3.2995 3.8996 3.2840 3.8699 3.2528 3.8131
4 5.2247 4.2520 5.2002 4.2199 5.1521 4.1585
5 8.2760 6.2383 8.2372 6.1915 8.1615 6.1015
6 12.0972 8.9993 12.0405 8.9318 11.9302 8.8021
7 16.6337 12.3380 16.5556 12.2455 16.4042 12.0676
8 21.8714 16.2115 21.7714 16.0899 21.5724 15.8563
9 27.8155 20.6086 27.6849 20.4541 27.4319 20.1517
10 34.4549 25.5258 34.2951 25.3345 33.9817 24.9666

Table 14. Variation of natural frequencies (Hz) of 1st Type with C-F and C-SS boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 21.1529 9.4714 20.9911 9.3990 20.6817 9.2604
2 7.0938 3.1359 7.0396 3.0981 6.9357 3.0565
3 3.9006 2.4436 3.8710 2.4253 3.8136 2.3928
4 4.2529 3.8694 4.2210 3.8405 4.1590 3.7898
5 6.2389 6.1293 6.1922 6.0834 6.1018 6.0034
6 8.9998 8.9593 8.9323 8.8922 8.8024 8.7755
7 12.3383 12.3190 12.2458 12.2267 12.0678 12.0664
8 16.2117 16.2001 16.0902 16.0786 15.8564 15.8680
9 20.6088 20.6003 20.4543 20.4459 20.1572 20.1780
10 25.5260 25.5189 25.3346 25.3276 24.4967 24.9959

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.

Table 15. Variation of natural frequencies (Hz) of 2ndType with SS-SS and C-C boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n SS-SS C-C SS-SS C-C SS-SS C-C
1 12.678 20.895 12.753 21.054 12.896 21.378
2 4.1767 7.0094 4.2010 7.0626 4.2487 7.1711
3 3.2651 3.8502 3.2817 3.8793 3.3156 3.9387
4 5.1712 4.1898 5.1956 4.2212 5.2454 4.2853
5 8.1929 6.1432 8.2312 6.1890 8.3089 6.2829
6 11.976 8.8611 12.032 8.9272 12.145 9.0625
7 16.468 12.148 16.545 12.238 16.701 12.424
8 21.657 15.962 21.758 16.081 21.962 16.324
9 27.540 20.291 27.669 20.472 27.928 20.752
10 34.115 25.133 34.275 25.320 34.597 25.704

Table 16. Variation of natural frequencies (Hz) of 2ndType with C-F and CC-SS boundary condition (L=20, h=0.002, R=1, m=1).

  N=0.5 N=1 N=5
n C-F C-SS C-F C-SS C-F C-SS
1 20.895 9.3688 21.054 9.4399 21.378 9.5852
2 7.0089 3.0861 7.0621 3.1095 7.1709 3.1577
3 3.8492 2.4076 3.8783 2.4256 3.9382 2.4632
4 4.1889 3.8091 4.2202 3.8374 4.2848 3.8961
5 6.1426 6.0341 6.1883 6.0790 6.2825 6.1715
6 8.8607 8.8208 8.9267 8.8865 9.0623 9.0213
7 12.147 12.123 12.238 12.219 12.424 12.404
8 15.961 15.950 16.080 16.069 16.324 16.312
9 20.291 20.282 20.442 20.434 20.752 20.743
10 25.138 25.125 25.320 25.313 25.704 25.696

4. Conclusions

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.[4] 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.


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