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In the present study, vibration analysis of a three-layered cylindrical shell is performed whose inner and outer layers are composed of functionally graded materials whereas the middle one is assumed to be of isotropic material. This formation of a cylindrical shell influences stiffness modulii and the resultant material properties. The shell problem is formulated from the constitutive relations of stresses and strains with the displacement deformations and they are taken from Love’s thin shell theory. This problem is transformed into the integral form by evaluating the expressions for the strain and kinetic energies of the shell. Rayleigh-Ritz method is employed to solve the shell dynamic equations. Vibration characteristics of these cylindrical shells are investigated for a number of physical parameters and configurations of the fabrication of shells. The axial modal dependence is approximated by the characteristic beam functions that satisfy the boundary conditions. Results evaluated, show good agreement with the open literature.

Yamanouchi et al. [^{th} century, gave the first linear shell theory based on Krichhoff’s hypothesis for plates. Arnold and Warburton [

Most of functionally graded materials are used in high temperature and possess temperature dependent proper-

ties. The material property

where

the constituent materials. The material properties

where

tion, the extensional, coupling and bending stiffness are modified in three layers. Let

the inner and outer constituent materials of the FGM layers used to fabricate the three layered FGM cylindrical shell with middle layer of isotropic material. For a functionally graded material layers, consisting of two mate-

rials _{ }, volume fraction is written for an effective material property as:

where

where

cylindrical shell. Also

Consider a cylindrical shell as shown in the

thickness of the cylindrical shell. The orthogonal coordinates system

face of the shell. The

radial directions are represented by

lindrical shell, three dimensional problems are converted in to two dimensional by applying plane stress condition. The constitutive relation of stress and strain of a thin cylindrical shell is given by Hook’s law as:

where

vector and the strain vectors are defined as:

where

-plane. Similarly

So the relation (5) can be expressed as:

For isotropic materials the reduced stiffness

where

where

For a thin cylindrical shell the force and moment resultants are defined as:

where

plies:

where

and

where

where

The coupling stiffness

where

where

By substituting

A number of shell theories have arisen and are used. Among these theories however the Love’s shell theory is considered to be the first theory about shells and all other shell theories were derived from the Love’s shell theory by amending some physical terms. The strain-displacement and the curvature-displacement relations which are adopted from Love’s [

By substituting these values of strain displacement and curvature displacement from equations (24) and (25) in equation (23), we obtain the strain energy equation in the form of displacement functions

The Lagrangian energy functional

The energy variation methods i.e., Rayleigh Ritz and Galerkin methods are the most frequently used ones to analyze the shell vibrational behavior. In the Rayleigh Ritz method, the energy variational functional is minimized with respect to the coefficients of an approximating series representing the displacement deformations. Many researchers such as Sewall and Naumann [

The expressions for the modal displacement deformations are presumed in the form of product of functions of space and time variables. This leads to a system of ordinary differential equations of three unknown functions of the axial space variable. Different types of functions are chosen to approximate the axial modal dependence. These functions satisfy the boundary conditions. Well-known functions are beam functions, Ritz polynomial functions, orthogonal polynomials and Fourier series of the circular functions. The expression for modal displacement deformations are assumed as:

In the axial, circumferential and radial directions respectively, the coefficients

where

The geometric boundary conditions for clamped, free and simply supported boundary conditions can be expressed mathematically in terms of characteristic beam function

Clamped boundary condition

Free boundary condition

Simply supported boundary condition

On substituting the expressions for the deformation displacements

where

To derive the shell frequency equation, the energy functional is extremized with respect to the vibration amplitudes: A, B and C, resulting in three homogenous linear following equations:

By re-arranging equation (35), the shell frequency equation is written in the eigenvalue form as:

where the expressions for the terms

A number of comparison of the results for isotropic and FGM cylindrical shells are presented to verify the validity, efficiency and accuracy of the present approach. The present analysis is carried out by using the energy variational procedure viz: Rayleigh-Ritz method. This method is based on the principle of minimization of energy. The numerical results for the following three frequently encountered sets of boundary conditions are evaluated to check the validity, efficiency and accuracy of the present technique.

Boundary Conditions | Values for | ||
---|---|---|---|

SS-SS | 1 | ||

C-C | |||

F-F | |||

C-SS | |||

C-F | |||

F-SS |

・ Simply supported-simply supported (SS-SS)

・ Clamped-clamped (C-C)

・ Clamped-free (C-F)

In

those ones evaluated by Swaddiwudhipong [

In

In

A three-layered FGM cylindrical shell whose cross section is shown in

Swaddiwudhipong [ | Present | ||
---|---|---|---|

20 | 1 | 0.016101 | 0.016101 |

2 | 0.005453 | 0.005450 | |

3 | 0.005042 | 0.005034 | |

4 | 0.008534 | 0.008525 | |

5 | ------ | 0.013623 | |

0.25 | 1 | 0.951993 | 0.951976 |

2 | 0.934461 | 0.934342 | |

3 | 0.906732 | 0.906435 | |

4 | 0.87076 | 0.870196 | |

5 | ------ | 0.827882 |

n | Joseph and Haim[ | Present |
---|---|---|

3 | 0.1030 | 0.1097 |

4 | 0.0681 | 0.0715 |

5 | 0.0515 | 0.0532 |

6 | 0.0475 | 0.0482 |

7 | 0.0528 | 0.0529 |

8 | 0.0639 | 0.0638 |

9 | 0.0788 | 0.0785 |

10 | 0.0964 | 0.0960 |

Sewall and Nauman [ | Present | |||||
---|---|---|---|---|---|---|

2 | --- | --- | --- | 354 | 1500.6 | 2346 |

3 | 155.0 157.0 | --- | --- | 182 | 912 | 1715 |

4 | 107.0 | --- | --- | 114 | 588 | 1248 |

5 | 89.0 91.0 | 341.0 | --- | 95 | 407 | 925 |

6 | 102.0 | 276.0 | --- | 106 | 306 | 707 |

7 | 130.0 | 240.0 | --- | 134 | 256 | 561 |

8 | 166.0 | 227.0 231.0 | --- | 172 | 243 | 469 |

9 | 208.0 | 246.0 | 400.0 | 217 | 259 | 418 |

10 | 260.0 | 281.0 | --- | 267 | 294 | 403 |

11 | 317.0 | 337.0 | 409.0 412.0 | 324 | 341 | 415 |

12 | 374.0 | 393.0 396.0 | --- | 385 | 398 | 449 |

Loy et al. [ | Present | |||||
---|---|---|---|---|---|---|

0.5 | 1.0 | 5.0 | 0.5 | 1.0 | 5.0 | |

Type I Shell | ||||||

1 | 13.321 | 13.211 | 12.998 | 13.331 | 13.210 | 12.988 |

2 | 4.5168 | 4.480 | 4.4068 | 4.5175 | 4.4790 | 4.4045 |

3 | 4.1911 | 4.1569 | 4.0891 | 4.1909 | 4.1560 | 4.0883 |

4 | 7.0972 | 7.0384 | 6.9251 | 7.0965 | 7.0371 | 6.9244 |

5 | 11.336 | 11.241 | 11.061 | 11.3350 | 11.2404 | 11.0603 |

Type II Shell | ||||||

1 | 13.154 | 13.3210 | 13.526 | 13.1545 | 13.3210 | 13.5052 |

2 | 4.4550 | 4.5114 | 4.5836 | 4.4550 | 4.5115 | 4.5759 |

3 | 4.1309 | 4.1827 | 4.2536 | 4.1308 | 4.1829 | 4.2450 |

4 | 7.0076 | 7.0903 | 7.2085 | 7.0034 | 7.0909 | 7.1945 |

5 | 11.189 | 11.3293 | 11.516 | 11.1896 | 11.3305 | 11.4944 |

fluencing the shell vibrations characteristics. In this study the Poisson’s ratio is assumed to be constant for functionally graded materials whereas the Young’s modulus is a function of the intrinsic thickness variable

where

By keeping isotropic material at the middle layer and by the variation of the constituents in the FGM layer as shown in

Material properties for inner and outer FGM layers of the cylindrical shell vary from

These results conclude that the material properties vary smoothly and continuously of constituent materials

ly. Similar response of the material properties is seen in the inverse direction. Variation of volume fractions

faces at the inner and outer FGM layers respectively of the shell are sketched in _{ }of constituent material

material

respectively. The middle layer is made up with some isotropic material, whose thickness span over the interval

Now to study the influence of power law exponent

layers,

Type of Shell | Inner FGM layer | Isotropic Layer | Outer FGM layer |
---|---|---|---|

Shell I | Nickel-Zarconia | Stainless Steel | Nickel-Zarconia |

Shell II | Nickel-Zarconia | Stainless Steel | Zarconia-Nickel |

Shell III | Zarconia-Nickel | Stainless Steel | Nickel-Zarconia |

Shell IV | Zarconia-Nickel | Stainless Steel | Zarconia-Nickel |

and outer FGM layers. Similar but opposite behaviour of the FGM constituents is observed at outer surface of both the layers. For the thickness interval

In this section variation of natural frequencies (Hz) for four types of shell described in the above table will be discussed. The boundary conditions are taken to be simply supported-simply supported (SS-SS), clamped-clamed (C-C) and clamped-free (C-F).

In table 7, the variation of natural frequencies (Hz) for three sets of boundary conditions i.e. simply supported-simply supported (SS-SS), clamped-clamped (C-C) and clamped-free(C-F) are studied against the circumferential wave number

SS-SS | C-C | C-F | |
---|---|---|---|

Type I Shell | |||

1 | 16.1932 | 34.6722 | 6.1921 |

2 | 5.4819 | 12.1477 | 2.5145 |

3 | 5.0470 | 7.2610 | 4.4855 |

4 | 8.5343 | 9.0512 | 8.4259 |

5 | 13.6324 | 13.7689 | 13.6005 |

Type II Shell | |||

1 | 16.0239 | 34.3095 | 6.1273 |

2 | 5.4174 | 12.0174 | 2.4726 |

3 | 4.9318 | 7.1420 | 4.3684 |

4 | 8.3093 | 8.8288 | 8.2003 |

5 | 13.2676 | 13.4052 | 13.2358 |

Type III Shell | |||

1 | 16.0239 | 34.3095 | 6.1273 |

2 | 5.4275 | 12.0219 | 2.4946 |

3 | 5.0199 | 7.2029 | 4.4673 |

4 | 8.5006 | 9.0089 | 8.3940 |

5 | 13.5807 | 13.7149 | 13.5494 |

Type IV Shell | |||

1 | 15.8527 | 33.9430 | 6.0619 |

2 | 5.3624 | 11.8903 | 2.4524 |

3 | 4.9040 | 7.0828 | 4.3497 |

4 | 8.2747 | 8.7854 | 8.1676 |

5 | 13.2145 | 13.3496 | 13.1832 |

taining its minimum value it begins to increase with the circumferential wave number

In

From tables it is observed that the natural frequency (Hz) for all three boundary conditions decreases with the increase of length to radius ratio. It is also seen that the frequencies related to the clamped-clamped boundary conditions are greater than the simply supported-simply supported and clamped-free boundary conditions and the difference between the frequencies for all boundary conditions becomes very negligible at

In

SS-SS | C-C | C-F | |
---|---|---|---|

Type I Shell | |||

0.5 | 676.7750 | 720.7210 | 442.8185 |

1.0 | 356.1130 | 438.4916 | 171.9393 |

5.0 | 23.9693 | 48.6353 | 12.0134 |

10 | 10.1874 | 15.5551 | 8.6910 |

50 | 8.4088 | 8.4222 | 8.4054 |

Type II Shell | |||

0.5 | 669.6757 | 713.1185 | 438.1774 |

1.0 | 352.3808 | 433.8956 | 170.1311 |

5.0 | 23.6691 | 48.1026 | 11.7902 |

10 | 9.9667 | 15.3182 | 8.4666 |

50 | 8.1833 | 8.1968 | 8.1799 |

Type III Shell | |||

0.5 | 669.7055 | 713.2100 | 438.1912 |

1.0 | 352.3917 | 433.9097 | 170.1452 |

5.0 | 23.7391 | 48.1366 | 11.9279 |

10 | 10.1277 | 15.4230 | 8.6548 |

50 | 8.3771 | 8.3903 | 8.3738 |

Type IV Shell | |||

0.5 | 662.5305 | 705.5265 | 433.5006 |

1.0 | 348.6196 | 429.2646 | 168.3177 |

5.0 | 23.4359 | 47.5983 | 11.7031 |

10 | 9.9057 | 15.1841 | 8.4294 |

50 | 8.1507 | 8.1640 | 8.1474 |

SS-SS | C-C | C-F | |
---|---|---|---|

Type I Shell | |||

0.001 | 4.4440 | 5.3762 | 4.2386 |

0.005 | 21.0814 | 21.2855 | 21.0286 |

0.01 | 42.0892 | 42.1729 | 42.0464 |

0.03 | 126.1878 | 126.1485 | 126.1154 |

0.05 | 210.2552 | 210.1509 | 210.1423 |

Type II Shell | |||

0.001 | 4.3324 | 5.2663 | 4.1260 |

0.005 | 20.5174 | 20.7230 | 20.4646 |

0.01 | 40.9607 | 41.0458 | 40.9183 |

0.03 | 122.8027 | 122.7657 | 122.7320 |

0.05 | 204.6163 | 204.5155 | 204.5063 |

Type III Shell | |||

0.001 | 4.4242 | 5.3420 | 4.2222 |

0.005 | 21.0015 | 21.2019 | 20.9494 |

0.01 | 41.9306 | 42.0126 | 41.8882 |

0.03 | 125.7130 | 125.6734 | 125.6409 |

0.05 | 209.4636 | 209.3594 | 209.3512 |

Type IV Shell | |||

0.001 | 4.3121 | 5.2314 | 4.1091 |

0.005 | 20.4352 | 20.6372 | 20.3832 |

0.01 | 40.7977 | 40.8810 | 40.7558 |

0.03 | 122.3149 | 122.2775 | 122.2445 |

0.05 | 203.8029 | 203.7022 | 203.6933 |

natural frequencies for all three boundary conditions are very close to each other. For every boundary condition, it is also observed from the tables that the natural frequencies increase as the thickness-to-radius ratio increase.

where