The vibration characteristics of ring stiffened cylindrical shells are analyzed. These shells are assumed to be structured from functionally graded materials (FGM) and are stiffened with isotropic rings. The problem is formulated by coupling the expressions for strain and kinetic energies of a circular cylindrical shell with those for rings. The Lagrangian function is framed by taking difference of strain and kinetic energies. The Rayleigh-Ritz approach is employed to obtain shell dynamical equations. The axial model dependence is approximated by characteristic beam functions that satisfy the boundary conditions. The validity and efficiency of the present technique are verified by comparisons of present results with the previous ones determined by other researchers.
Circular cylindrical shells stiffened by rings are widely used in many structural applications such as airplanes, marine crafts, pressure vessels, silos, core barrels of pressurized water reactors, submarine hulls, offshore drilling rings, and construction buildings. Usually cylinders are stiffened by rings or strings to increase the stiffness and strength, reduce the weight structure to be designed. In designing these shells, it is vital to know their resonant frequencies because excessive vibrations can lead to fatigue rupture.
First proper shell theory was proposed by Love [
There is bulk of studies on isotropic homogeneous materials and the studies on new invented composite materials such as functionally graded materials (FGMs) have been carried out by many researchers. In this material compositions and functions are varying continuously from one side to the other side. For example, one side may have high mechanical strength and the other side may have high thermal resistant property; thus, there are “two aspects” in one material. In FGMs, change of compositions is continuous and it does not come out from simply “bonding individual substances.” This generates boundaries among the bonded ones. They are all considered by mechanically and chemically. Koizumi [
Although an extensive amount of research work has been carried out to study the vibration characteristics of isotropic as well as composite cylindrical shells, there is no evidence of work on vibration of functionally graded cylindrical shells stiffener with ring support. The present study is concerned with analysis of the vibration characteristics of functionally graded cylindrical shell stiffened with isotropic ring on the outer surface of the shell. The Rayleigh-Ritz method is used to formulate the shell eigen-frequency equation.
Consider a circular thin shell of uniform thickness
Cylindrical shell with Ring-Stiffeners.
For a thin cylindrical shell, plane stress condition is assumed and the constitutive relation for a thin cylindrical shell is given by
Stress vector and strain vector are defined as
The Rayleigh-Ritz approach is employed to analyze vibrational behaviour of stiffened functionally graded circular cylindrical shells. The use of Ritz method provides a rapid convergence and excellent accuracy. The following displacement functions
Using the expressions for the functions
The FGMs are much advanced materials and are used in engineering science and technology. Material properties of an FGM are the functions of the temperature and the position. These properties of a constituent material are managed by a volume fraction. If
Numerical technique known as the Rayleigh-Ritz method has been employed to study the vibration characteristics of functionally graded circular cylindrical shells with ring- stiffeners. To confirm the efficiency and validity of the present method, the frequencies of cylindrical shells with and without ring-stiffeners are compared with those values found in the literature.
In Table
Comparison of frequency parameters
|
Loy et al. [ |
Present |
---|---|---|
1 | 0.016102 | 0.016101 |
2 | 0.009387 | 0.009381 |
3 | 0.022108 | 0.022105 |
4 | 0.042096 | 0.042095 |
5 | 0.068008 | 0.068008 |
6 | 0.099730 | 0.099730 |
7 | 0.137239 | 0.137240 |
8 | 0.180527 | 0.180529 |
9 | 0.229594 | 0.229596 |
10 | 0.284435 | 0.284439 |
The often-cited Galletly [
Comparison studies of frequency parameter
Stiffener's depth-to-width ratio |
Circumferential wave no. |
Swaddiwudhipong et al. [ |
Present analysis |
---|---|---|---|
1.3314 | 2 | 0.08472 | 0.08503 |
1.3314 | 3 | 0.06711 | 0.07049 |
1.3314 | 4 | 0.1072 | 0.11493 |
1.3314 | 5 | 0.1714 | 0.18394 |
2.6628 | 2 | 0.08372 | 0.08591 |
2.6628 | 3 | 0.1095 | 0.12269 |
2.6628 | 4 | 0.2037 | 0.2296 |
2.6628 | 5 | 0.3271 | 0.37226 |
3.9942 | 2 | 0.08788 | 0.09371 |
3.9942 | 3 | 0.1592 | 0.18444 |
3.9942 | 4 | 0.3049 | 0.35343 |
3.9942 | 5 | 0.4598 | 0.57304 |
Comparison studies of frequency parameter
Stiffener's depth-to-width ratio |
Circumferential wave no. |
Swaddiwudhipong et al. [ |
Present analysis |
---|---|---|---|
1.3314 | 2 | 0.08282 | 0.0876 |
1.3314 | 3 | 0.05992 | 0.0771 |
1.3314 | 4 | 0.08641 | 0.1221 |
1.3314 | 5 | 0.1346 | 0.1918 |
2.6628 | 2 | 0.08141 | 0.0933 |
2.6628 | 3 | 0.08826 | 0.1380 |
2.6628 | 4 | 0.1536 | 0.2499 |
2.6628 | 5 | 0.2459 | 0.4000 |
3.9942 | 2 | 0.08425 | 0.1076 |
3.9942 | 3 | 0.1280 | 0.2128 |
3.9942 | 4 | 0.2363 | 0.3970 |
3.9942 | 5 | 0.3807 | 0.6377 |
Frequency study is analyzed for 14 evenly spaced eccentric ring stiffeners externally and internally in Tables
The functionally graded material considered here is composed of stainless steel and nickel. The variations of natural frequencies of a functionally graded cylindrical shell are compared with those determined by Naeem et al. [
Properties of materials.
Coefficients | Stainless steel | Nickel | ||||
---|---|---|---|---|---|---|
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| |
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0.3262 | 8166 |
|
0.3100 | 8900 |
|
0 | 0 | 0 | 0 | 0 | 0 |
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0 |
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0 | 0 |
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0 |
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0 | 0 |
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0 | 0 | 0 | 0 | 0 | 0 |
| ||||||
|
|
8166 |
|
0.3100 | 8900 |
Natural frequencies (Hz) determined by the present method for a simply supported functionally graded cylindrical shell are presented in Tables
Comparison of natural frequencies (Hz) for type I FG cylindrical shell with simply-supported boundary conditions. (
|
Naeem et al. [ |
Present | ||||
---|---|---|---|---|---|---|
|
|
|
|
|
|
|
1 | 13.321 | 13.211 | 12.933 | 13.321 | 13.211 | 12.932 |
2 | 4.5162 | 4.4794 | 4.3829 | 4.5164 | 4.4796 | 4.3830 |
3 | 4.1903 | 4.1562 | 4.0646 | 4.1909 | 4.1564 | 4.0641 |
4 | 7.0967 | 7.0379 | 6.8851 | 7.0976 | 7.0381 | 6.8836 |
5 | 11.335 | 11.241 | 10.998 | 11.336 | 11.241 | 10.996 |
6 | 16.594 | 16.455 | 16.101 | 16.595 | 16.455 | 16.097 |
7 | 22.826 | 22.635 | 22.148 | 22.828 | 22.635 | 22.143 |
8 | 30.023 | 29.771 | 29.132 | 30.025 | 29.771 | 29.125 |
9 | 38.181 | 37.862 | 37.048 | 38.185 | 37.862 | 37.039 |
10 | 47.301 | 46.905 | 45.897 | 47.305 | 46.905 | 45.886 |
Comparison of natural frequencies (Hz) for type II FG cylindrical shell with simply-supported boundary conditions. (
|
Naeem et al. [ |
Present | ||||
---|---|---|---|---|---|---|
|
|
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|
|
|
1 | 13.103 | 13.211 | 13.505 | 13.103 | 13.211 | 13.505 |
2 | 4.4382 | 4.4742 | 4.5759 | 4.4378 | 4.4739 | 4.5757 |
3 | 4.1152 | 4.1486 | 4.2451 | 4.1144 | 4.1481 | 4.2454 |
4 | 6.9754 | 7.0330 | 7.1943 | 6.9744 | 7.0327 | 7.1956 |
5 | 11.145 | 11.238 | 11.494 | 11.143 | 11.237 | 11.496 |
6 | 16.317 | 16.453 | 16.453 | 16.315 | 16.452 | 16.830 |
7 | 22.447 | 22.633 | 23.147 | 22.444 | 22.633 | 23.152 |
8 | 29.524 | 29.770 | 30.446 | 29.521 | 29.770 | 30.453 |
9 | 37.548 | 37.861 | 38.720 | 37.544 | 37.861 | 38.729 |
10 | 46.517 | 46.904 | 47.968 | 46.511 | 46.904 | 47.979 |
In Table
Tables
Variation of natural frequencies against circumferential wave number
|
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
1 | 26.831 | 25.864 | 26.497 | 26.421 | 26.335 | 26.176 | 26.018 | 25.922 | 25.894 |
2 | 30.397 | 29.289 | 30.014 | 29.926 | 29.828 | 29.645 | 29.466 | 29.355 | 29.323 |
3 | 47.192 | 45.494 | 46.603 | 46.468 | 46.317 | 46.037 | 45.762 | 45.594 | 45.545 |
4 | 82.416 | 79.471 | 81.393 | 81.158 | 80.896 | 80.411 | 79.935 | 79.643 | 79.559 |
5 | 131.10 | 126.42 | 129.47 | 129.10 | 128.68 | 127.91 | 127.16 | 126.69 | 126.56 |
6 | 191.52 | 184.69 | 189.14 | 188.60 | 187.99 | 186.86 | 185.76 | 185.09 | 184.89 |
7 | 263.21 | 253.82 | 259.59 | 259.20 | 258.37 | 256.82 | 255.30 | 254.37 | 254.11 |
8 | 346.02 | 333.69 | 341.74 | 340.75 | 339.66 | 337.62 | 335.63 | 334.41 | 334.o6 |
9 | 439.88 | 424.20 | 434.43 | 433.18 | 431.79 | 429.21 | 426.68 | 425.12 | 424.68 |
10 | 544.74 | 525.34 | 538.01 | 536.46 | 534.73 | 531.53 | 528.40 | 526.48 | 525.93 |
Variation of natural frequencies against circumferential wave number
|
|
|
|
|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
1 | 14.715 | 14.025 | 14.436 | 14.421 | 14.360 | 14.246 | 14.134 | 14.066 | 14.046 |
2 | 35.517 | 33.764 | 34.836 | 34.768 | 34.618 | 34.342 | 34.068 | 33.885 | 33.828 |
3 | 96.256 | 91.453 | 94.364 | 94.184 | 93.772 | 93.017 | 92.275 | 91.782 | 91.626 |
4 | 183.17 | 174.01 | 179.56 | 179.21 | 178.43 | 176.99 | 175.57 | 174.64 | 174.34 |
5 | 295.50 | 280.72 | 289.66 | 289.11 | 287.84 | 285.52 | 283.24 | 281.73 | 281.25 |
6 | 433.06 | 411.40 | 424.51 | 423.70 | 421.84 | 418.44 | 415.10 | 412.88 | 412.18 |
7 | 595.76 | 565.96 | 584.00 | 582.89 | 580.33 | 575.65 | 571.05 | 568.00 | 567.04 |
8 | 783.53 | 744.35 | 768.06 | 766.60 | 763.24 | 757.08 | 751.04 | 747.03 | 745.76 |
9 | 996.33 | 946.51 | 976.66 | 974.80 | 970.53 | 962.70 | 955.02 | 949.92 | 948.31 |
10 | 1234.1 | 1172.4 | 1209.7 | 1207.46 | 1202.17 | 1192.5 | 1182.9 | 1176.6 | 1174.6 |
Tables
Variation of natural frequencies against circumferential wave number
|
|
|
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|
|
|
|
|
|
---|---|---|---|---|---|---|---|---|---|
1 | 26.831 | 25.864 | 26.176 | 26.251 | 26.335 | 26.498 | 26.663 | 26.768 | 26.798 |
2 | 30.397 | 29.289 | 29.647 | 29.732 | 29.830 | 30.016 | 30.204 | 30.324 | 30.359 |
3 | 47.192 | 45.494 | 46.044 | 46.176 | 46.325 | 46.611 | 46.897 | 47.082 | 47.135 |
4 | 82.416 | 79.471 | 80.426 | 80.655 | 80.914 | 81.408 | 81.904 | 82.225 | 82.317 |
5 | 131.10 | 126.42 | 127.94 | 128.30 | 128.71 | 129.50 | 130.28 | 130.80 | 130.94 |
6 | 191.52 | 184.69 | 186.90 | 187.43 | 188.03 | 189.18 | 190.33 | 191.08 | 191.29 |
7 | 263.21 | 253.82 | 256.87 | 257.60 | 258.42 | 260.00 | 261.58 | 262.60 | 262.90 |
8 | 346.02 | 333.69 | 337.69 | 338.65 | 339.73 | 341.80 | 343.88 | 345.22 | 345.61 |
9 | 439.88 | 424.20 | 429.29 | 430.51 | 431.89 | 434.52 | 437.15 | 438.87 | 439.36 |
10 | 544.74 | 525.34 | 531.64 | 533.15 | 534.85 | 538.11 | 541.37 | 543.49 | 544.10 |
Variation of natural frequencies against circumferential wave number
|
|
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|
|
|
---|---|---|---|---|---|---|---|---|---|
1 | 14.714 | 14.025 | 14.245 | 14.298 | 14.359 | 14.475 | 14.594 | 14.669 | 14.691 |
2 | 35.517 | 33.764 | 34.329 | 34.461 | 34.610 | 34.891 | 35.178 | 35.383 | 35.446 |
3 | 96.256 | 91.453 | 93.018 | 93.383 | 93.790 | 94.559 | 95.344 | 95.892 | 96.063 |
4 | 183.17 | 174.01 | 177.00 | 177.70 | 178.47 | 179.94 | 181.43 | 182.48 | 182.80 |
5 | 295.50 | 280.72 | 285.54 | 286.66 | 287.92 | 290.28 | 292.70 | 294.38 | 294.90 |
6 | 433.06 | 411.40 | 418.47 | 420.12 | 421.96 | 425.42 | 428.96 | 431.42 | 432.19 |
7 | 595.76 | 565.96 | 575.69 | 577.96 | 580.49 | 585.25 | 590.12 | 593.50 | 594.57 |
8 | 783.53 | 744.35 | 757.14 | 760.12 | 763.44 | 769.71 | 776.11 | 780.56 | 781.96 |
9 | 996.33 | 946.51 | 962.78 | 966.56 | 970.79 | 978.76 | 986.89 | 992.55 | 994.33 |
10 | 1234.1 | 1172.4 | 1192.5 | 1197.2 | 1202.5 | 1212.3 | 1222.4 | 1229.4 | 1231.6 |
Comparing the frequencies in Tables
On the other hand, for
Tables
Variation of natural frequencies against circumferential wave number
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---|---|---|---|---|---|---|---|---|---|
1 | 25.274 | 24.348 | 24.955 | 24.881 | 24.799 | 24.646 | 24.496 | 24.403 | 24.377 |
2 | 29.526 | 28.439 | 29.151 | 29.064 | 28.968 | 28.788 | 28.613 | 28.504 | 28.473 |
3 | 45.740 | 44.073 | 45.161 | 45.029 | 44.881 | 44.605 | 44.337 | 44.171 | 44.124 |
4 | 79.445 | 76.570 | 78.446 | 78.217 | 77.962 | 77.487 | 77.024 | 76.739 | 76.658 |
5 | 126.21 | 121.65 | 124.63 | 124.26 | 123.86 | 123.10 | 122.37 | 121.92 | 121.79 |
6 | 184.32 | 177.66 | 182.00 | 181.47 | 180.88 | 179.78 | 178.71 | 178.05 | 177.86 |
7 | 253.29 | 244.15 | 250.11 | 249.38 | 248.57 | 247.06 | 245.59 | 244.68 | 244.42 |
8 | 332.97 | 320.96 | 328.79 | 327.83 | 326.77 | 324.79 | 322.85 | 321.66 | 321.32 |
9 | 423.29 | 408.03 | 417.98 | 416.77 | 415.41 | 412.90 | 410.43 | 408.92 | 408.49 |
10 | 524.21 | 505.31 | 517.64 | 516.14 | 514.46 | 511.34 | 508.29 | 506.42 | 505.88 |
Variation of natural frequencies against circumferential wave number
|
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|
---|---|---|---|---|---|---|---|---|---|
1 | 14.482 | 13.800 | 14.206 | 14.192 | 14.131 | 14.019 | 13.908 | 13.841 | 13.821 |
2 | 35.176 | 33.434 | 34.499 | 34.433 | 34.283 | 34.009 | 33.737 | 33.555 | 33.498 |
3 | 95.883 | 91.090 | 93.995 | 93.817 | 93.405 | 92.652 | 91.911 | 91.419 | 91.263 |
4 | 182.65 | 173.50 | 179.04 | 178.70 | 177.91 | 176.48 | 175.07 | 174.13 | 173.83 |
5 | 294.72 | 279.96 | 288.89 | 288.35 | 287.08 | 284.76 | 282.48 | 280.97 | 280.49 |
6 | 431.95 | 410.31 | 423.41 | 422.60 | 420.75 | 417.35 | 414.01 | 411.79 | 411.09 |
7 | 594.24 | 564.47 | 582.49 | 581.38 | 578.83 | 574.15 | 569.56 | 566.51 | 565.55 |
8 | 781.54 | 742.40 | 766.08 | 764.63 | 761.27 | 755.12 | 749.08 | 745.07 | 743.81 |
9 | 993.81 | 944.04 | 974.15 | 972.31 | 968.04 | 960.22 | 952.54 | 947.45 | 945.84 |
10 | 1231.0 | 1169.4 | 1206.6 | 1204.4 | 1199.1 | 1189.4 | 1179.9 | 1173.6 | 1171.6 |
Tables
Variation of natural frequencies against circumferential wave number
|
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|
---|---|---|---|---|---|---|---|---|---|
1 | 25.274 | 24.348 | 24.646 | 24.718 | 24.799 | 24.955 | 25.113 | 25.214 | 25.243 |
2 | 29.526 | 28.439 | 28.790 | 28.874 | 28.970 | 29.152 | 29.338 | 29.455 | 29.490 |
3 | 45.740 | 44.073 | 44.613 | 44.742 | 44.888 | 45.168 | 45.451 | 45.632 | 45.684 |
4 | 79.445 | 76.570 | 77.502 | 77.726 | 77.978 | 78.461 | 78.945 | 79.259 | 79.349 |
5 | 126.21 | 121.65 | 123.12 | 123.48 | 123.88 | 124.65 | 125.42 | 125.91 | 126.06 |
6 | 184.32 | 177.66 | 179.81 | 180.34 | 180.92 | 182.04 | 183.16 | 183.89 | 184.09 |
7 | 253.29 | 244.15 | 247.11 | 247.82 | 248.63 | 250.16 | 251.70 | 252.70 | 252.99 |
8 | 332.97 | 320.96 | 324.85 | 325.79 | 326.85 | 328.86 | 330.88 | 332.20 | 332.57 |
9 | 423.29 | 408.03 | 412.97 | 414.16 | 415.51 | 418.07 | 420.64 | 422.30 | 422.78 |
10 | 524.21 | 505.31 | 511.44 | 512.91 | 514.57 | 517.74 | 520.92 | 522.99 | 523.58 |
Variation of natural frequencies against circumferential wave number
|
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---|---|---|---|---|---|---|---|---|---|
1 | 14.482 | 13.800 | 14.018 | 14.071 | 14.130 | 14.245 | 14.362 | 14.436 | 14.458 |
2 | 35.176 | 33.434 | 33.995 | 34.126 | 34.274 | 34.553 | 34.839 | 35.043 | 35.105 |
3 | 95.883 | 91.090 | 92.651 | 93.015 | 93.421 | 94.188 | 94.972 | 95.519 | 95.691 |
4 | 182.65 | 173.50 | 176.49 | 177.18 | 177.96 | 179.42 | 180.91 | 181.95 | 182.28 |
5 | 294.72 | 279.96 | 284.78 | 285.90 | 287.15 | 289.51 | 291.92 | 293.60 | 294.13 |
6 | 431.95 | 410.31 | 417.37 | 419.02 | 420.85 | 424.31 | 427.85 | 430.31 | 431.08 |
7 | 594.24 | 564.47 | 574.19 | 576.45 | 578.98 | 583.74 | 588.60 | 591.98 | 593.05 |
8 | 781.54 | 742.40 | 755.17 | 758.14 | 761.47 | 767.73 | 774.12 | 778.57 | 779.97 |
9 | 993.81 | 944.04 | 960.28 | 964.06 | 968.29 | 976.24 | 984.38 | 990.03 | 991.81 |
10 | 1231.0 | 1169.4 | 1189.5 | 1194.2 | 1199.4 | 1209.3 | 1219.3 | 1226.3 | 1228.5 |
Comparing the frequencies in Tables
Tables
Variation of natural frequencies against
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0.2 | 20 | 2875.5 | 2866.8 | 2872.5 | 2871.8 | 2871.1 | 2869.6 | 2868.2 | 2867.3 | 2867.1 |
0.5 | 15 | 1614.0 | 1602.9 | 1610.2 | 1609.4 | 1608.4 | 1606.5 | 1604.7 | 1603.5 | 1603.3 |
1.0 | 11 | 900.77 | 890.59 | 897.30 | 896.49 | 895.59 | 893.89 | 892.22 | 891.20 | 890.90 |
2.0 | 8 | 488.92 | 480.57 | 486.06 | 485.40 | 484.66 | 483.27 | 481.91 | 481.07 | 480.82 |
5.0 | 5 | 181.84 | 177.15 | 180.23 | 179.85 | 179.44 | 178.66 | 177.90 | 177.43 | 177.29 |
10 | 4 | 100.74 | 97.538 | 99.630 | 99.380 | 99.095 | 98.566 | 98.046 | 97.727 | 97.636 |
20 | 3 | 45.740 | 44.073 | 45.162 | 45.029 | 44.881 | 44.605 | 44.336 | 44.171 | 44.124 |
50 | 2 | 13.408 | 12.874 | 13.222 | 13.179 | 13.132 | 13.044 | 12.958 | 12.905 | 12.890 |
100 | 1 | 2.5497 | 2.4455 | 2.5135 | 2.5052 | 2.4960 | 2.4788 | 2.4620 | 2.4516 | 2.4486 |
Variation of natural frequencies against
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0.2 | 20 | 2875.5 | 2866.8 | 2869.3 | 2870.5 | 2871.2 | 2872.7 | 2874.1 | 2875.0 | 2875.2 |
0.5 | 15 | 1614.0 | 1602.9 | 1606.3 | 1607.6 | 1608.5 | 1610.4 | 1612.2 | 1613.4 | 1613.7 |
1.0 | 11 | 900.77 | 890.59 | 893.66 | 894.81 | 895.71 | 897.41 | 899.07 | 900.15 | 900.45 |
2.0 | 8 | 488.92 | 480.57 | 483.07 | 484.00 | 484.74 | 486.14 | 487.51 | 488.40 | 488.65 |
5.0 | 5 | 181.84 | 177.15 | 178.52 | 179.06 | 179.47 | 180.26 | 181.04 | 181.54 | 181.68 |
10 | 4 | 100.74 | 97.538 | 98.430 | 98.833 | 99.115 | 99.652 | 100.19 | 100.54 | 100.64 |
20 | 3 | 45.740 | 44.073 | 44.505 | 44.742 | 44.888 | 45.168 | 45.450 | 45.631 | 45.684 |
50 | 2 | 13.408 | 12.874 | 12.994 | 13.086 | 13.133 | 13.223 | 13.314 | 13.373 | 13.390 |
100 | 1 | 2.5497 | 2.4455 | 2.4788 | 2.4868 | 2.4959 | 2.5135 | 2.5314 | 2.5427 | 2.5461 |
Tables
Variation of fundamental natural frequencies against
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0.001 | 3 | 52.567 | 50.896 | 51.992 | 51.859 | 51.711 | 51.434 | 51.163 | 50.995 |
0.005 | 2 | 24.993 | 23.985 | 24.643 | 24.563 | 24.473 | 24.307 | 24.145 | 24.045 |
0.007 | 2 | 23.114 | 22.162 | 22.783 | 22.707 | 22.623 | 22.466 | 22.313 | 22.218 |
0.01 | 2 | 21.393 | 20.493 | 21.081 | 21.000 | 20.929 | 20.782 | 20.637 | 20.547 |
0.02 | 1 | 15.812 | 15.090 | 15.561 | 15.504 | 15.440 | 15.321 | 15.204 | 15.133 |
0.03 | 1 | 15.096 | 14.396 | 15.060 | 14.798 | 14.736 | 14.621 | 14.507 | 14.438 |
0.04 | 1 | 14.717 | 14.028 | 14.472 | 14.424 | 14.362 | 14.249 | 14.137 | 14.069 |
0.05 | 1 | 14.482 | 13.800 | 14.245 | 14.191 | 14.131 | 14.019 | 13.908 | 13.841 |
Variation of fundamental natural frequencies against
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0.001 | 3 | 52.567 | 50.896 | 51.439 | 51.569 | 51.716 | 51.996 | 52.279 | 52.459 |
0.005 | 2 | 24.993 | 23.985 | 24.309 | 24.387 | 24.475 | 24.645 | 24.816 | 24.927 |
0.007 | 2 | 23.114 | 22.162 | 22.467 | 22.541 | 22.624 | 22.784 | 22.945 | 23.050 |
0.01 | 2 | 21.393 | 20.493 | 20.782 | 20.851 | 20.930 | 21.081 | 21.232 | 21.333 |
0.02 | 1 | 15.812 | 15.090 | 15.321 | 15.376 | 15.440 | 15.561 | 15.685 | 15.764 |
0.03 | 1 | 15.096 | 14.396 | 14.620 | 14.674 | 14.735 | 14.853 | 14.974 | 15.050 |
0.04 | 1 | 14.717 | 14.028 | 14.248 | 14.301 | 14.361 | 14.477 | 14.596 | 14.671 |
0.05 | 1 | 14.482 | 13.800 | 14.018 | 14.070 | 14.130 | 14.245 | 14.362 | 14.436 |
The frequency characteristics of FGM cylindrical shells with ring-stiffeners are similar to those for homogeneous isotropic cylindrical shells. Other interesting frequency characteristics are also observed in the FGM cylindrical shells. These characteristics arise when the constituent volume fractions and the configurations of the constituent materials in the functionally graded cylindrical shells are varied in the thickness direction.
In this study, the Rayleigh-Ritz approach has been employed to analyze the vibration characteristics of functionally graded circular cylindrical shells with ring-stiffeners of different materials. The axial model dependence has been approximated by the characteristic beam functions. Sander’s thin shell theory of first order has been used to perform the vibration analysis. From the vibration results of cylindrical shells with identical and evenly spaced ring-stiffeners, it is found that stiffeners placed eccentrically are more effective than concentric ones. The study is carried out for isotropic as well as two types of functionally graded cylindrical shell with and without ring stiffeners where the configurations of the constituent materials in the functionally graded cylindrical shells are varied by the volume fraction law. One is termed as Type I FG cylindrical shell and has properties that vary continuously from Nickel on its inner surface and Stainless Steel on its outer surface. The other is termed as a Type II FG cylindrical shell and has properties that vary continuously from Stainless Steel on its inner surface and Nickel on its outer surface. A validation of the analysis has been carried out by comparing results with those found in literature and a good agreement has been observed among the results evaluated by different shell theories and numerical approaches. It is seen that the variations of natural frequency of FGM circular cylinders are similar to that of isotropic ones. The frequency is influenced by the volume fraction law exponents. It decreases or increases with N depending upon the order of constituent material in FGM shells. For the Type I and Type II FG cylindrical shells the natural frequencies for all values of N lie between those for a Stainless Steel and Nickel cylindrical shells. For
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