The paper proposes a three-dimensional elastic analysis of the free vibration problem of one-layered spherical, cylindrical, and flat panels. The exact solution is developed for the differential equations of equilibrium written in orthogonal curvilinear coordinates for the free vibrations of simply supported structures. These equations consider an exact geometry for shells without simplifications. The main novelty is the possibility of a general formulation for different geometries. The equations written in general orthogonal curvilinear coordinates allow the analysis of spherical shell panels and they automatically degenerate into cylindrical shell panel, cylindrical closed shell, and plate cases. Results are proposed for isotropic and orthotropic structures. An exhaustive overview is given of the vibration modes for a number of thickness ratios, imposed wave numbers, geometries, embedded materials, and angles of orthotropy. These results can also be used as reference solutions to validate two-dimensional models for plates and shells in both analytical and numerical form (e.g., closed solutions, finite element method, differential quadrature method, and global collocation method).
Exact solutions for the three-dimensional analysis of plates and shells have been developed by several researchers. In particular, three-dimensional static and dynamic analysis of cylinders and cylindrical/spherical shells has received considerable attention in the past few years. The development of plate and shell elements highlights the importance of research on the design of these geometries because they are fundamental for the analysis of general structures in the engineering field. For this reason, plate and shell elements need accurate validation. Three-dimensional exact solutions allow such validations and checks to be made, and they also give further details about three-dimensional effects and their importance. In the literature, works about exact three-dimensional solutions analyze separately the various geometries and they do not give a general overview of plate and shell elements. The present paper aims to fill this gap by proposing a general formulation for the equations of motion in orthogonal curvilinear coordinates that is valid for square and rectangular plates, cylindrical shell panels, spherical shell panels, and cylinders. A general overview is given for those readers interested in both plate and shell analysis. This paper exactly solves the equations of motion in general curvilinear orthogonal coordinates including an exact geometry for shell structures without simplifications. To the best of the present author’s knowledge, this is the first time that this solution is proposed by means of the exponential matrix method for the three-dimensional elastic free vibration analysis of plates and shells.
The next paragraph discusses the most relevant works about three-dimensional shell analysis. Chen et al. [
Three-dimensional analysis of plates is usually performed by using simpler equations that do not allow the analysis of shell geometries. Demasi [
The revision of the literature about three-dimensional analysis of shells and plates demonstrates that there are a variety of interesting works concerning plate or shell geometry, and they include static and dynamic analysis, functionally graded, composite and piezoelectric materials, displacement, and mixed models. The present work gives a general formulation for all the geometries (square and rectangular plates, cylindrical and spherical shell panels, and cylindrical closed shells). The equations of motion for the dynamic case are written in general orthogonal curvilinear coordinates by using an exact geometry for shells. The system of second order differential equations is reduced to a system of first order differential equations, and afterwards it is exactly solved by using the exponential matrix method. Similar approaches have been used in [
A shell is a three-dimensional body bounded by two closely spaced curved surfaces where one dimension (the distance between the two surfaces) is small in comparison with the other two dimensions in the plane directions. The middle surface of the shell is the locus of points which lie midway between these surfaces. The distance between the surfaces measured along the normal to the middle surface is the
Notation and reference system for shells.
In this work, we will focus on shells with constant radii of curvature (e.g., cylindrical and spherical geometries). The geometrical relations written for shells with constant radii of curvature are a particular case of the strain-displacement equations of three-dimensional theory of elasticity in orthogonal curvilinear coordinates (see also [
Geometrical relations (see (
The three differential equations of equilibrium written for the case of free vibration analysis of spherical shells with constant radii of curvature
Equation (
The system of second order differential equations can be reduced to a system of first order differential equations by using the method described in [
In the case of plate geometry coefficients
Thickness coordinates for plates and shells.
In the case of shell geometry matrices
The structures are simply supported and free stresses at the top and at the bottom; this feature means:
The free vibration problem means finding the nontrivial solution of
A certain number of circular frequencies
It is possible to find
The three-dimensional exact solution proposed has been validated by means of a comparison with three assessments given in the literature for a one-layered isotropic plate [
Geometries considered for the assessments and benchmarks: (a) square plate, (b) rectangular plate, (c) cylindrical shell panel, (d) cylinder, and (e) spherical shell panel.
The assessment proposed in Vel and Batra [
Assessment 1. Simply supported aluminum square plate with Poisson ratio
Mode | ||||||
---|---|---|---|---|---|---|
I | II | III | IV | V | VI | |
|
||||||
Present 3D ( |
5.3370 | 27.554 | 46.493 | 154.39 | 168.09 | 268.36 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 192.91 | 197.98 | 329.35 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.21 | 201.87 | 308.34 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.77 | 201.34 | 358.23 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.77 | 201.34 | 357.42 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.71 | 201.27 | 355.40 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.77 | 201.34 | 351.90 |
Present 3D ( |
5.7769 | 27.554 | 46.502 | 196.77 | 201.34 | 357.42 |
3D [ |
5.7769 | 27.554 | 46.503 | 196.77 | 201.34 | 357.42 |
3D [ |
5.7769 | 27.554 | 46.502 | 196.77 | 201.34 | 357.42 |
|
||||||
| ||||||
Present 3D ( |
/ | 8.7132 | 14.182 | 17.512 | / | / |
Present 3D ( |
4.6598 | 8.7132 | 14.463 | 20.987 | 23.999 | 29.925 |
Present 3D ( |
4.6582 | 8.7132 | 14.463 | 21.383 | 25.009 | 31.399 |
Present 3D ( |
4.6581 | 8.7132 | 14.463 | 21.343 | 24.830 | 34.049 |
Present 3D ( |
4.6581 | 8.7132 | 14.463 | 21.343 | 24.830 | 33.982 |
Present 3D ( |
4.6580 | 8.7132 | 14.463 | 21.337 | 24.821 | 33.939 |
Present 3D ( |
4.6581 | 8.7132 | 14.463 | 21.343 | 24.830 | 33.923 |
Present 3D ( |
4.6581 | 8.7132 | 14.463 | 21.343 | 24.830 | 33.982 |
3D [ |
4.6582 | 8.7132 | 14.463 | 21.343 | 24.830 | 33.982 |
3D [ |
4.6582 | 8.7134 | 14.462 | 21.343 | 24.830 | 33.982 |
The assessment proposed by Khalili et al. [
Assessment 2 (part I). Simply supported aluminum cylinder with Poisson ratio
|
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---|---|---|---|---|
|
|
|
|
|
|
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Present 3D ( |
0.03724 | 0.02323 | 0.02175 | 0.02846 |
Present 3D ( |
0.03730 | 0.02360 | 0.02464 | 0.03687 |
Present 3D ( |
0.03730 | 0.02359 | 0.02463 | 0.03686 |
Present 3D ( |
0.03730 | 0.02359 | 0.02462 | 0.03686 |
3D [ |
0.03730 | 0.02359 | 0.02462 | 0.03686 |
|
||||
|
||||
Present 3D ( |
0.05769 | 0.04772 | 0.04229 | 0.04272 |
Present 3D ( |
0.05853 | 0.04979 | 0.04790 | 0.05546 |
Present 3D ( |
0.05853 | 0.04978 | 0.04789 | 0.05545 |
Present 3D ( |
0.05853 | 0.04978 | 0.04789 | 0.05545 |
3D [ |
0.05853 | 0.04978 | 0.04789 | 0.05545 |
|
||||
|
||||
Present 3D ( |
0.08017 | 0.07732 | 0.07308 | 0.06667 |
Present 3D ( |
0.10057 | 0.10234 | 0.10688 | 0.11528 |
Present 3D ( |
0.10057 | 0.10234 | 0.10688 | 0.11528 |
Present 3D ( |
0.10057 | 0.10234 | 0.10688 | 0.11528 |
3D [ |
0.10057 | 0.10234 | 0.10688 | 0.11528 |
|
||||
|
||||
Present 3D ( |
0.48156 | 0.48619 | 0.49378 | 0.50419 |
Present 3D ( |
0.27489 | 0.27848 | 0.28446 | 0.29286 |
Present 3D ( |
0.27491 | 0.27849 | 0.28447 | 0.29287 |
Present 3D ( |
0.27491 | 0.27849 | 0.28447 | 0.29287 |
3D [ |
0.27491 | 0.27849 | 0.28447 | 0.29287 |
Assessment 2 (part II). Simply supported aluminum cylinder with Poisson ratio
|
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---|---|---|---|---|
|
|
|
|
|
|
||||
Present 3D ( |
0.05613 | 0.03607 | 0.03241 | 0.02570 |
Present 3D ( |
0.05652 | 0.03930 | 0.04998 | 0.07822 |
Present 3D ( |
0.05652 | 0.03929 | 0.04996 | 0.07821 |
Present 3D ( |
0.05652 | 0.03929 | 0.04996 | 0.07821 |
3D [ |
0.05652 | 0.03929 | 0.04996 | 0.07821 |
|
||||
|
||||
Present 3D ( |
0.08833 | 0.07237 | 0.05717 | 0.01965 |
Present 3D ( |
0.09403 | 0.08546 | 0.09094 | 0.11205 |
Present 3D ( |
0.09402 | 0.08545 | 0.09093 | 0.11205 |
Present 3D ( |
0.09402 | 0.08545 | 0.09093 | 0.11205 |
3D [ |
0.09402 | 0.08545 | 0.09093 | 0.11205 |
|
||||
|
||||
Present 3D ( |
0.36506 | 0.37946 | 0.40165 | 0.43001 |
Present 3D ( |
0.18895 | 0.19467 | 0.20616 | 0.22449 |
Present 3D ( |
0.18894 | 0.19467 | 0.20616 | 0.22450 |
Present 3D ( |
0.18894 | 0.19467 | 0.20616 | 0.22450 |
3D [ |
0.18894 | 0.19467 | 0.20616 | 0.22450 |
|
||||
|
||||
Present 3D ( |
0.72230 | 0.72911 | 0.74032 | 0.75570 |
Present 3D ( |
0.50335 | 0.50934 | 0.51931 | 0.53321 |
Present 3D ( |
0.50338 | 0.50937 | 0.51934 | 0.53325 |
Present 3D ( |
0.50338 | 0.50937 | 0.51934 | 0.53325 |
3D [ |
0.50338 | 0.50937 | 0.51934 | 0.53325 |
The assessment proposed by Soldatos and Ye [
Assessment 3. Simply supported orthotropic cylinder with thickness ratio
Mode | ||||||
---|---|---|---|---|---|---|
I | II | III | I | II | III | |
( |
( |
|||||
Present 3D ( |
1.02407 | 2.55671 | 4.66395 | 0.20406 | 0.49393 | 0.91564 |
Present 3D ( |
0.32274 | 0.41827 | 3.29830 | 0.30493 | 0.81346 | 7.51166 |
Present 3D ( |
0.99999 | 1.29840 | 1.85418 | 0.95308 | 1.00000 | 3.04558 |
Present 3D ( |
1.00000 | 1.29714 | 2.48260 | 0.95308 | 1.00000 | 3.25249 |
Present 3D ( |
1.00000 | 1.29714 | 2.48260 | 0.95308 | 1.00000 | 3.25249 |
3D [ |
1.00000 | 1.29730 | 2.48260 | 0.95307 | 1.00000 | 3.25250 |
All the new benchmarks proposed in Section
Ten different benchmarks are proposed to show a complete overview of the free vibration analysis of one-layered plates and shells. Five different geometries are taken into account in Figure
Benchmark 1. Simply supported isotropic square plate. First three vibration modes in term of no-dimensional circular frequency
|
100 | 50 | 10 | 5 |
---|---|---|---|---|
( |
||||
I mode | 2.9861 | 2.9846 | 2.9360 | 2.8012 |
II mode | 194.83 | 97.417 | 19.483 | 9.7417 |
III mode | 329.33 | 164.66 | 32.908 | 16.415 |
|
||||
( |
||||
I mode | 2.9861 | 2.9846 | 2.9360 | 2.8012 |
II mode | 194.83 | 97.417 | 19.483 | 9.7417 |
III mode | 329.33 | 164.66 | 32.908 | 16.415 |
|
||||
( |
||||
I mode | 5.9713 | 5.9650 | 5.7769 | 5.3036 |
II mode | 275.54 | 137.77 | 27.554 | 13.777 |
III mode | 465.73 | 232.86 | 46.502 | 23.136 |
|
||||
( |
||||
I mode | 14.920 | 14.881 | 13.805 | 11.645 |
II mode | 435.66 | 217.83 | 43.566 | 21.783 |
III mode | 736.37 | 368.14 | 73.350 | 36.157 |
|
||||
( |
||||
I mode | 14.920 | 14.881 | 13.805 | 11.645 |
II mode | 435.66 | 217.83 | 43.566 | 21.783 |
III mode | 736.37 | 368.14 | 73.350 | 36.157 |
|
||||
( |
||||
I mode | 23.860 | 23.761 | 21.214 | 16.882 |
II mode | 551.07 | 275.54 | 55.107 | 27.554 |
III mode | 931.42 | 465.63 | 92.545 | 45.071 |
Benchmark 2. Simply supported orthotropic square plate. First three vibration modes in term of no-dimensional circular frequency
|
100 | 50 | 10 | 5 | 100 | 50 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
|
|
|||||||
I mode | 2.8553 | 2.8537 | 2.8058 | 2.6731 | 10.007 | 9.9712 | 9.0075 | 7.2059 |
II mode | 227.77 | 113.88 | 22.777 | 11.388 | 227.77 | 113.88 | 22.777 | 11.388 |
III mode | 314.89 | 157.44 | 31.457 | 15.679 | 1104.7 | 552.35 | 110.41 | 46.861 |
|
||||||||
|
|
|||||||
I mode | 10.007 | 9.9712 | 9.0075 | 7.2059 | 2.8553 | 2.8537 | 2.8058 | 2.6731 |
II mode | 227.77 | 113.88 | 22.777 | 11.388 | 227.77 | 113.88 | 22.777 | 11.388 |
III mode | 1104.7 | 552.35 | 110.41 | 46.861 | 314.89 | 157.44 | 31.457 | 15.679 |
|
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|
|
|||||||
I mode | 11.367 | 11.325 | 10.226 | 8.2183 | 11.367 | 11.325 | 10.226 | 8.2183 |
II mode | 382.03 | 191.01 | 38.169 | 19.032 | 382.03 | 191.01 | 38.169 | 19.032 |
III mode | 1130.2 | 565.09 | 112.94 | 49.636 | 1130.2 | 565.09 | 112.94 | 49.636 |
|
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|
|
|||||||
I mode | 17.719 | 17.648 | 15.817 | 12.656 | 41.011 | 40.436 | 29.614 | 19.220 |
II mode | 652.40 | 326.17 | 65.004 | 32.100 | 549.29 | 274.64 | 54.882 | 27.380 |
III mode | 1204.5 | 602.22 | 120.31 | 56.667 | 2222.2 | 1111.0 | 190.33 | 53.344 |
|
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|
|
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I mode | 41.011 | 40.436 | 29.614 | 19.220 | 17.719 | 17.648 | 15.817 | 12.656 |
II mode | 549.29 | 274.64 | 54.882 | 27.380 | 652.40 | 326.17 | 65.004 | 32.100 |
III mode | 2222.2 | 1111.0 | 190.33 | 53.344 | 1204.5 | 602.22 | 120.31 | 56.667 |
|
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|
|
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I mode | 45.302 | 44.664 | 32.873 | 21.768 | 45.302 | 44.664 | 32.873 | 21.768 |
II mode | 764.05 | 381.98 | 76.128 | 37.625 | 764.05 | 381.98 | 76.128 | 37.625 |
III mode | 2260.4 | 1130.1 | 198.54 | 59.836 | 2260.4 | 1130.1 | 198.54 | 59.836 |
Benchmark 3. Simply supported isotropic rectangular plate. First three vibration modes in terms of no-dimensional circular frequency
|
100 | 50 | 10 | 5 |
---|---|---|---|---|
|
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I mode | 0.3318 | 0.3318 | 0.3312 | 0.3293 |
II mode | 64.944 | 32.472 | 6.4944 | 3.2472 |
III mode | 109.78 | 54.888 | 10.977 | 5.4870 |
|
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|
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I mode | 2.9861 | 2.9846 | 2.9360 | 2.8012 |
II mode | 194.83 | 97.417 | 19.483 | 9.7417 |
III mode | 329.33 | 164.66 | 32.908 | 16.415 |
|
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|
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I mode | 3.3179 | 3.3159 | 3.2562 | 3.0923 |
II mode | 205.37 | 102.69 | 20.537 | 10.269 |
III mode | 347.14 | 173.56 | 34.685 | 17.297 |
|
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|
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I mode | 4.3130 | 4.3097 | 4.2099 | 3.9450 |
II mode | 234.16 | 117.08 | 23.416 | 11.708 |
III mode | 395.80 | 197.89 | 39.537 | 19.699 |
|
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|
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I mode | 12.270 | 12.243 | 11.497 | 9.9094 |
II mode | 395.04 | 197.52 | 39.504 | 19.752 |
III mode | 667.72 | 333.83 | 66.559 | 32.908 |
|
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|
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I mode | 13.264 | 13.233 | 12.369 | 10.572 |
II mode | 410.74 | 205.37 | 41.074 | 20.537 |
III mode | 694.26 | 347.10 | 69.186 | 34.170 |
Benchmark 4. Simply supported orthotropic rectangular plate. First three vibration modes in terms of no-dimensional circular frequency
|
100 | 50 | 10 | 5 | 100 | 50 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
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I mode | 0.3173 | 0.3173 | 0.3167 | 0.3148 | 1.1130 | 1.1126 | 1.0987 | 1.0586 |
II mode | 75.922 | 37.961 | 7.5922 | 3.7961 | 75.922 | 37.961 | 7.5922 | 3.7961 |
III mode | 104.98 | 52.483 | 10.495 | 5.2459 | 368.24 | 184.12 | 36.822 | 18.408 |
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I mode | 10.007 | 9.9712 | 9.0075 | 7.2059 | 2.8553 | 2.8537 | 2.8058 | 2.6731 |
II mode | 227.77 | 113.88 | 22.777 | 11.388 | 227.77 | 113.68 | 22.777 | 11.388 |
III mode | 1104.7 | 552.35 | 110.41 | 46.861 | 314.89 | 157.44 | 31.457 | 15.679 |
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I mode | 10.127 | 10.091 | 9.1126 | 7.2879 | 3.4236 | 3.4213 | 3.3511 | 3.1611 |
II mode | 249.70 | 124.85 | 24.967 | 12.480 | 312.77 | 156.38 | 31.251 | 15.586 |
III mode | 1107.6 | 553.78 | 110.69 | 47.184 | 441.11 | 220.55 | 44.099 | 22.031 |
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I mode | 10.538 | 10.500 | 9.4774 | 7.5865 | 6.1047 | 6.0947 | 5.8032 | 5.1319 |
II mode | 306.16 | 153.08 | 30.604 | 15.284 | 341.88 | 170.94 | 34.157 | 17.031 |
III mode | 1116.1 | 558.04 | 111.54 | 48.130 | 774.35 | 387.17 | 77.407 | 38.652 |
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I mode | 40.001 | 39.445 | 28.902 | 18.694 | 11.868 | 11.841 | 11.093 | 9.5179 |
II mode | 466.89 | 233.44 | 46.684 | 23.337 | 560.57 | 280.28 | 56.021 | 27.943 |
III mode | 2210.8 | 1105.3 | 187.77 | 51.133 | 656.59 | 328.31 | 65.435 | 32.318 |
|
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I mode | 40.364 | 39.801 | 29.152 | 18.872 | 13.685 | 13.649 | 12.644 | 10.634 |
II mode | 499.39 | 249.69 | 49.921 | 24.938 | 625.52 | 312.73 | 62.343 | 30.811 |
III mode | 2215.1 | 1107.5 | 188.74 | 51.982 | 882.21 | 441.09 | 88.126 | 43.879 |
Benchmark 5. Simply supported isotropic cylindrical shell panel. First three vibration modes in terms of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 |
---|---|---|---|---|
|
||||
I mode | 1067.1 | 106.71 | 10.683 | 5.3415 |
II mode | 1068.3 | 106.83 | 10.690 | 5.3727 |
III mode | 1855.7 | 185.57 | 18.557 | 9.2797 |
|
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|
||||
I mode | 2.5478 | 2.5184 | 2.4853 | 2.3948 |
II mode | 2040.3 | 204.03 | 20.411 | 10.216 |
III mode | 3635.3 | 363.51 | 36.229 | 17.917 |
|
||||
|
||||
I mode | 222.72 | 22.503 | 3.8758 | 3.2156 |
II mode | 2364.5 | 236.45 | 23.656 | 11.842 |
III mode | 4017.4 | 401.73 | 40.060 | 19.846 |
|
||||
|
||||
I mode | 552.89 | 55.574 | 7.7851 | 5.7590 |
II mode | 3031.5 | 303.16 | 30.328 | 15.181 |
III mode | 5048.7 | 504.86 | 50.367 | 24.983 |
|
||||
|
||||
I mode | 70.382 | 14.052 | 11.533 | 10.021 |
II mode | 4230.8 | 423.08 | 42.325 | 21.174 |
III mode | 7214.2 | 721.38 | 71.724 | 35.120 |
|
||||
|
||||
I mode | 232.90 | 27.451 | 13.793 | 11.643 |
II mode | 4639.2 | 463.92 | 46.414 | 23.226 |
III mode | 7847.2 | 784.68 | 78.020 | 38.182 |
Benchmark 6. Simply supported orthotropic cylindrical shell panel. First three vibration modes in term of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 | 1000 | 100 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
|
|
|||||||
I mode | 1248.9 | 124.89 | 12.489 | 6.2444 | 1096.5 | 109.67 | 11.281 | 6.0108 |
II mode | 1721.6 | 172.16 | 17.230 | 8.6360 | 1248.9 | 124.89 | 12.489 | 6.2444 |
III mode | 3858.5 | 385.85 | 38.590 | 19.293 | 6057.8 | 605.78 | 60.570 | 30.272 |
|
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|
|
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I mode | 8.4450 | 8.4392 | 7.6484 | 6.1872 | 2.3272 | 2.4079 | 2.3754 | 2.2863 |
II mode | 2385.2 | 238.52 | 23.861 | 11.941 | 2385.2 | 238.52 | 23.861 | 11.943 |
III mode | 12194 | 1219.4 | 121.48 | 51.358 | 3476.0 | 347.58 | 34.633 | 17.114 |
|
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|
|
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I mode | 230.37 | 24.568 | 8.0953 | 6.3709 | 352.48 | 35.490 | 5.3604 | 4.1227 |
II mode | 2939.9 | 294.00 | 29.454 | 14.782 | 3588.6 | 358.85 | 35.788 | 17.735 |
III mode | 12257 | 1225.7 | 122.10 | 52.106 | 6559.0 | 655.90 | 65.574 | 32.756 |
|
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|
|
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I mode | 641.70 | 64.887 | 10.877 | 6.8670 | 644.27 | 65.496 | 12.428 | 9.0851 |
II mode | 4155.7 | 415.58 | 41.604 | 20.824 | 4164.3 | 416.43 | 41.609 | 20.709 |
III mode | 12445 | 1244.4 | 123.94 | 54.240 | 12370 | 1237.0 | 123.62 | 54.652 |
|
||||||||
|
|
|||||||
I mode | 79.744 | 39.112 | 28.352 | 18.587 | 160.90 | 20.150 | 11.545 | 9.9019 |
II mode | 5062.8 | 506.29 | 50.681 | 25.371 | 6476.9 | 647.66 | 64.461 | 31.673 |
III mode | 23492 | 2349.1 | 205.91 | 56.149 | 7983.5 | 798.33 | 79.684 | 39.571 |
|
||||||||
|
|
|||||||
I mode | 243.62 | 46.084 | 29.034 | 19.077 | 364.43 | 40.611 | 16.527 | 13.080 |
II mode | 5849.5 | 584.96 | 58.566 | 29.293 | 6951.1 | 695.08 | 69.132 | 33.884 |
III mode | 23598 | 2359.6 | 208.24 | 58.169 | 13115 | 1311.5 | 131.01 | 62.003 |
Benchmark 7. Simply supported isotropic cylinder. First three vibration modes in term of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 |
---|---|---|---|---|
( |
||||
I mode | 7691.7 | 769.17 | 76.917 | 38.459 |
II mode | 12342 | 1234.2 | 123.42 | 61.710 |
III mode | 41586 | 4158.6 | 416.30 | 208.81 |
|
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|
||||
I mode | 2445.9 | 244.59 | 24.492 | 12.295 |
II mode | 27270 | 2727.0 | 272.81 | 136.56 |
III mode | 59189 | 5918.8 | 591.08 | 294.28 |
|
||||
|
||||
I mode | 7712.1 | 771.23 | 77.252 | 38.820 |
II mode | 33375 | 3337.5 | 333.85 | 167.09 |
III mode | 61353 | 6135.3 | 612.74 | 305.15 |
|
||||
|
||||
I mode | 13536 | 1353.7 | 135.69 | 68.323 |
II mode | 40081 | 4008.1 | 400.89 | 200.56 |
III mode | 65490 | 6548.9 | 654.17 | 325.96 |
Benchmark 8. Simply supported orthotropic cylinder. First three vibration modes in terms of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 | 1000 | 100 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
|
|
|||||||
I mode | 8991.9 | 899.19 | 89.919 | 44.960 | 8991.9 | 899.19 | 89.919 | 44.959 |
II mode | 12402 | 1240.2 | 124.16 | 62.281 | 39097 | 3909.7 | 391.39 | 196.33 |
III mode | 138826 | 13883 | 1389.6 | 696.66 | 44039 | 4403.9 | 440.45 | 220.32 |
|
||||||||
|
|
|||||||
I mode | 2486.4 | 248.65 | 24.921 | 12.541 | 5295.1 | 529.53 | 53.084 | 26.738 |
II mode | 31697 | 3169.8 | 317.14 | 158.81 | 50390 | 5039.0 | 503.83 | 251.80 |
III mode | 196438 | 19643 | 1961.3 | 974.72 | 58023 | 5802.3 | 579.62 | 288.86 |
|
||||||||
|
|
|||||||
I mode | 8071.9 | 807.21 | 80.903 | 40.715 | 11839 | 1184.0 | 119.10 | 60.562 |
II mode | 38989 | 3898.9 | 390.16 | 195.44 | 56607 | 5660.7 | 565.47 | 281.81 |
III mode | 196773 | 19677 | 1964.6 | 976.14 | 92402 | 9240.2 | 924.00 | 461.97 |
|
||||||||
|
|
|||||||
I mode | 14640 | 1464.1 | 146.76 | 73.903 | 17656 | 1765.9 | 178.90 | 92.578 |
II mode | 48229 | 4823.0 | 482.66 | 241.82 | 58769 | 5876.8 | 587.41 | 293.24 |
III mode | 197336 | 19733 | 1970.0 | 978.55 | 134283 | 13428 | 1342.8 | 671.29 |
Benchmark 9. Simply supported isotropic spherical shell panel. First three vibration modes in terms of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 |
---|---|---|---|---|
|
||||
I mode | 999.01 | 99.924 | 10.213 | 5.3986 |
II mode | 2040.3 | 204.02 | 20.360 | 10.115 |
III mode | 3785.7 | 378.55 | 37.677 | 18.556 |
|
||||
|
||||
I mode | 999.01 | 99.924 | 10.213 | 5.3986 |
II mode | 2040.3 | 204.02 | 20.360 | 10.116 |
III mode | 3785.7 | 378.55 | 37.677 | 18.556 |
|
||||
|
||||
I mode | 1046.3 | 104.76 | 11.564 | 6.9554 |
II mode | 2885.4 | 288.53 | 28.793 | 14.302 |
III mode | 5111.6 | 511.13 | 50.821 | 24.937 |
|
||||
|
||||
I mode | 1076.3 | 108.52 | 16.906 | 12.400 |
II mode | 4562.2 | 456.21 | 45.524 | 22.598 |
III mode | 7857.9 | 785.72 | 77.933 | 37.847 |
|
||||
|
||||
I mode | 1076.3 | 108.52 | 16.906 | 12.400 |
II mode | 4562.2 | 456.21 | 45.524 | 22.598 |
III mode | 7857.9 | 785.72 | 77.933 | 37.847 |
|
||||
|
||||
I mode | 1084.1 | 110.77 | 23.273 | 17.513 |
II mode | 5770.8 | 577.07 | 57.581 | 28.565 |
III mode | 9869.7 | 986.87 | 97.645 | 46.869 |
Benchmark 10. Simply supported orthotropic spherical shell panel. First three vibration modes in terms of no-dimensional circular frequency
|
1000 | 100 | 10 | 5 | 1000 | 100 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
|
|
|||||||
I mode | 2385.2 | 238.51 | 23.802 | 11.825 | 1037.9 | 104.08 | 12.585 | 7.7554 |
II mode | 2902.0 | 290.20 | 29.026 | 14.513 | 2385.2 | 238.51 | 23.802 | 11.824 |
III mode | 4371.6 | 437.15 | 43.559 | 21.518 | 12223 | 1222.3 | 121.65 | 51.888 |
|
||||||||
|
|
|||||||
I mode | 1037.9 | 104.08 | 12.583 | 7.7554 | 2385.2 | 238.51 | 23.802 | 11.825 |
II mode | 2385.2 | 238.51 | 23.802 | 11.824 | 2902.0 | 290.20 | 29.026 | 14.513 |
III mode | 12223 | 1222.3 | 121.65 | 51.888 | 4371.6 | 437.15 | 43.559 | 21.518 |
|
||||||||
|
|
|||||||
I mode | 1209.7 | 121.32 | 14.754 | 9.1755 | 1209.7 | 121.32 | 14.754 | 9.1755 |
II mode | 4016.5 | 401.63 | 40.041 | 19.819 | 4016.5 | 401.63 | 40.041 | 19.819 |
III mode | 12456 | 1245.5 | 123.88 | 54.506 | 12456 | 1245.5 | 123.88 | 54.506 |
|
||||||||
|
|
|||||||
I mode | 1682.9 | 169.01 | 21.986 | 14.305 | 1114.0 | 117.90 | 30.685 | 19.645 |
II mode | 6838.7 | 683.84 | 68.035 | 33.373 | 5758.3 | 575.82 | 57.401 | 28.398 |
III mode | 13147 | 1314.6 | 130.57 | 60.847 | 23599 | 2359.8 | 208.50 | 58.572 |
|
||||||||
|
|
|||||||
I mode | 1114.0 | 117.90 | 30.685 | 19.645 | 1682.9 | 169.01 | 21.986 | 14.305 |
II mode | 5758.3 | 575.82 | 57.401 | 28.398 | 6838.7 | 683.84 | 68.035 | 33.373 |
III mode | 23599 | 2359.8 | 208.50 | 58.572 | 13147 | 1314.6 | 130.57 | 60.847 |
|
||||||||
|
|
|||||||
I mode | 1260.9 | 133.19 | 34.372 | 22.411 | 1260.9 | 133.19 | 34.372 | 22.411 |
II mode | 8009.4 | 800.89 | 79.625 | 39.004 | 8009.4 | 800.89 | 79.625 | 39.004 |
III mode | 23985 | 2398.3 | 216.62 | 65.349 | 23985 | 2398.3 | 216.62 | 65.349 |
Benchmark 1, simply supported isotropic square plate. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 2, simply supported orthotropic square plate
Benchmark 2, simply supported orthotropic square plate (
Benchmark 7, simply supported isotropic cylinder. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 8, simply supported orthotropic cylinder
Benchmark 8, simply supported orthotropic cylinder
Benchmark 9, simply supported isotropic spherical shell panel. First three vibration modes in terms of displacement components through the thickness for thickness ratio
Benchmark 10, simply supported orthotropic spherical shell panel
Benchmark 10, simply supported orthotropic spherical shell panel
Table
Table
Tables
Table
Tables
Tables
The parametric coefficient effects, in particular the
Effects of
|
1000 | 100 | 10 | 5 |
---|---|---|---|---|
|
||||
I mode | 1067.1 | 106.71 | 10.683 | 5.3415 |
II mode | 1068.3 | 106.83 | 10.690 | 5.3727 |
III mode | 1855.7 | 185.57 | 18.557 | 9.2797 |
I mode ( |
1067.1 | 106.71 | 10.686 | 5.3486 |
II mode ( |
1068.3 | 106.83 | 10.688 | 5.3684 |
III mode ( |
1855.7 | 185.56 | 18.550 | 9.2651 |
|
||||
|
||||
I mode | 2.5478 | 2.5184 | 2.4853 | 2.3948 |
II mode | 2040.3 | 204.03 | 20.411 | 10.216 |
III mode | 3635.3 | 363.51 | 36.229 | 17.917 |
I mode ( |
2.7747 | 2.7473 | 2.7029 | 2.5828 |
II mode ( |
2040.3 | 204.03 | 20.403 | 10.201 |
III mode ( |
3635.3 | 363.52 | 36.303 | 18.076 |
|
||||
|
||||
I mode | 222.72 | 22.503 | 3.8758 | 3.2156 |
II mode | 2364.5 | 236.45 | 23.656 | 11.842 |
III mode | 4017.4 | 401.73 | 40.060 | 19.846 |
I mode ( |
222.73 | 22.535 | 4.0440 | 3.3738 |
II mode ( |
2364.5 | 236.45 | 23.668 | 11.868 |
III mode ( |
4017.4 | 401.73 | 40.115 | 19.965 |
Effects of
|
1000 | 100 | 10 | 5 | 1000 | 100 | 10 | 5 |
---|---|---|---|---|---|---|---|---|
|
|
|||||||
I mode | 2385.2 | 238.51 | 23.802 | 11.825 | 1037.9 | 104.08 | 12.585 | 7.7554 |
II mode | 2902.0 | 290.20 | 29.026 | 14.513 | 2385.2 | 238.51 | 23.802 | 11.824 |
III mode | 4371.6 | 437.15 | 43.559 | 21.518 | 12223 | 1222.3 | 121.65 | 51.888 |
I mode ( |
2385.2 | 238.52 | 23.858 | 11.939 | 1037.9 | 104.09 | 12.602 | 7.7410 |
II mode ( |
2902.0 | 290.20 | 29.030 | 14.529 | 2385.2 | 238.52 | 23.856 | 11.934 |
III mode ( |
4371.6 | 437.15 | 43.623 | 21.667 | 12223 | 1222.3 | 122.04 | 51.743 |
|
||||||||
|
|
|||||||
I mode | 1037.9 | 104.08 | 12.583 | 7.7554 | 2385.2 | 238.51 | 23.802 | 11.825 |
II mode | 2385.2 | 238.51 | 23.802 | 11.824 | 2902.0 | 290.20 | 29.026 | 14.513 |
III mode | 12223 | 1222.3 | 121.65 | 51.888 | 4371.6 | 437.15 | 43.559 | 21.518 |
I mode ( |
1037.9 | 104.09 | 12.602 | 7.7410 | 2385.2 | 238.52 | 23.858 | 11.939 |
II mode ( |
2385.2 | 238.52 | 23.856 | 11.934 | 2902.0 | 290.20 | 29.030 | 14.529 |
III mode ( |
12223 | 1222.3 | 122.04 | 51.743 | 4371.6 | 437.15 | 43.623 | 21.667 |
|
||||||||
|
|
|||||||
I mode | 1209.7 | 121.32 | 14.754 | 9.1755 | 1209.7 | 121.32 | 14.754 | 9.1755 |
II mode | 4016.5 | 401.63 | 40.041 | 19.819 | 4016.5 | 401.63 | 40.041 | 19.819 |
III mode | 12456 | 1245.5 | 123.88 | 54.506 | 12456 | 1245.5 | 123.88 | 54.506 |
I mode ( |
1209.7 | 121.33 | 14.810 | 9.2169 | 1209.7 | 121.33 | 14.810 | 9.2169 |
II mode ( |
4016.5 | 401.64 | 40.137 | 20.022 | 4016.5 | 401.64 | 40.137 | 20.022 |
III mode ( |
12456 | 1245.6 | 124.29 | 54.312 | 12456 | 1245.6 | 124.29 | 54.312 |
The differential equations of equilibrium in orthogonal curvilinear coordinates for the free vibrations of simply supported structures have been exactly solved in three-dimensional form by using the exponential matrix method. The proposed general 3D formulation uses an exact geometry for shells, and it allows results for spherical, open cylindrical, closed cylindrical and flat panels to be obtained. The first three vibration modes have been investigated for several geometries, both isotropic and orthotropic layers, various thickness ratios, and imposed half-wave numbers. The modes plotted through the thickness make it possible to recognize the most complicated cases and these results will be useful benchmarks to validate future refined 2D models. The method is simple and intuitive and it will be extended to multilayered structures (also embedding functionally graded layers). This extension will give a global three-dimensional overview of the free vibration problem of one-layered and multilayered plates and shells. The present method has a fast convergence and it is not heavy from the computational point of view for each geometry (plates and shells). This feature is an important advantage with respect to other methods proposed in the literature that are valid only for a chosen geometry.
The author declares that there is no conflict of interests regarding the publication of this paper.