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.
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 by_{u}1(x, δ, t), u_{2}(x, δ, t) and u_{3}(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
where
and ∈_{1} , ∈_{2} and ∈_{12} denote strains which are related to the reference surface and k_{1}, k_{2} and k_{12} represent curvatures. Prime (′) indicates the matrix transposition. The entries of the matrix [C] are furnished as:
where x_{ij} represent the extensional, y_{ij}, coupling and z_{ij} , bending stiffness. (i, j = 1, 2 and 6). They are defined by the following formulas:
The reduced stiffness Q_{ij} for isotropic materials is stated as [10]:
Here E represents the Young’s modulus and μ denotes the Poisson ratio. The matrix y_{ij} = 0 for isotropic circular shaped CS and y_{ij} ≠ 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:
Following expressions are taken from [23] and written as
By substituting expressions (7) and (8) into equation (6), then ℑ becomes as:
The kinetic energy T for a CS is expressed as:
here time variable is denoted by t and the mass density per unit length is represented by ρ_{t} and is written as:
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:
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:
here
where the values of β_{i} , (i = 1, 2, 3, 4),depend upon the nature of the edge conditions and α_{m} denotes the roots of trigonometric or hyperbolic equations and the parameters, χ_{m} ′s depend on values of α_{m}.
Following dimensionless parameters are utilized to simplify the problem.
Now the expressions (13) are re-written as:
After making substitutions of the expressions (16) and their respective derivatives into the relations (9) and (10), ℑ_{max} and T_{max} are obtained using the principle of conservation of energy. By applying the principle of maximum energy, the Lagrange functional, Г_{max} takes the following form:
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 Г_{max} is extremized with regard to vibration coefficients: a_{m}, b_{m} and c_{m}, we obtain the following relations.
A system of homogeneous simultaneous equations in a_{m}, b_{m} and c_{m} is generated and is transformed into the eigenvalue problem as.
where [K] is the stiffness matrix and [M]represents the mass matrix and
and
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
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: E_{2}, μ_{2},ρ_{2} and for nickel material are: E_{1}, μ_{1},ρ_{1}. The thickness of each layer is presumed to be H/3. Then the actual material quantities for FGM layer are
given as:
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 E_{F} = E_{2}, μ_{F} = μ_{2} and ρ_{F} = ρ_{2} at z = −H/6 where for z = H/6 material properties becomeE_{F} = E_{1}, μ_{F} = μ_{1} and ρ_{F} = ρ_{1}. 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 V_{F} of a FGM is formulated as following function [24]
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:
where e be the standard irrational natural exponential base number. The material properties are written as:
Trigonometric volume fraction law for a FGM circular CS is stated as:
Here
The material parameters for FG cylindrical shell are written as:
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
Table 1
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
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
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 | Internal Layer | Central FGM Layer | External Layer |
---|---|---|---|
Type I | M_{A} | M_{B}/M_{C} | M_{A} |
Type II | M_{A} | M_{C}/M_{B} | M_{A} |
Table 5
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
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
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
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
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
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
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
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
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
Figure 4
Figure 5
Figure 6
Figure 7
Figure 8
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
where
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