The Numerical Investigation of Natural Vibrations of a Truncated Layered Conical Shell Filled with a Fluid

Sergey A. Bochkarev, Sergey V. Lekomtsev

Author details

Sergey A. Bochkarev
Institute of Continuous Media Mechanics, Perm, Russian Federation
Sergey V. Lekomtsev
Institute of Continuous Media Mechanics, Perm, Russian Federation

https://doi.org/10.15593/perm.mech/2025.1.01

The paper presents the numerical study of natural vibrations of truncated straight layered conical shells of rotations filled with an ideal compressible fluid. The behavior of the elastic structure and the fluid medium is described in the classical shell theory based on the Kirchhoff – Love hypotheses and Euler equations. The equations of motion of the shell together with the corresponding geometrical and physical relations are reduced to a system of ordinary differential equations with respect to new unknown values. The acoustic wave equation is written for the hydrodynamic pressure and transformed to a system of differential equations using the method of generalized differential quadrature. The solution of the formulated boundary value problem is developed using the Godunov orthogonal sweep method with a numerical integration of differential equations using the Runge – Kutta method of the fourth order. The calculation of the natural frequencies of vibrations is carried out using a combination of the stepwise procedure with the subsequent refinement of the values in the obtained range by the Muller method. The validity of the obtained results is confirmed by a comparison with the known numerical solutions. The dependences of the lowest vibration frequencies on the cone angle and ply angle of simply supported, rigidly clamped and cantilevered two-layer and three-layer composite conical shells are analyzed. The study allows us to assess the possibility of changing the lowest frequencies and the corresponding mode shapes of the vibration in relation to the preset combination of the cone angle, boundary conditions on the edges, layup scheme and ply angle of the composite shells. An extensive series of calculations has revealed the existence of configurations, for which the lowest frequencies exceed the values corresponding values of the equivalent layered circular cylindrical shell.

Keywords

classical shell theorystraight conical shelllayered materialGodunov's orthogonal sweep methodideal compressible fluidgeneralized differential quadrature methodnatural vibrations.

ЧИСЛЕННОЕ ИССЛЕДОВАНИЕ СОБСТВЕННЫХ КОЛЕБАНИЙ СЛОИСТЫХ УСЕЧЕННЫХ КОНИЧЕСКИХ ОБОЛОЧЕК, ЗАПОЛНЕННЫХ ЖИДКОСТЬЮ

Представлены результаты численных исследований собственных колебаний усеченных прямых слоистых конических оболочек вращения, полностью заполненных идеальной сжимаемой жидкостью. Поведение упругой конструкции и жидкой среды описывается в рамках классической теории оболочек, основанной на гипотезах Кирхгофа – Лява, и уравнений Эйлера. Уравнения движения оболочки совместно с соответствующими геометрическими и физическими соотношениями сводятся к системе обыкновенных дифференциальных уравнений относительно новых неизвестных. Акустическое волновое уравнение, записанное относительно гидродинамического давления, преобразуется к системе дифференциальных уравнений с помощью метода обобщенных дифференциальных квадратур. Решение сформулированной краевой задачи осуществляется методом ортогональной прогонки Годунова с численным интегрированием дифференциальных уравнений методом Рунге – Кутты четвертого порядка точности. Для вычисления собственных частот колебаний используется сочетание пошаговой процедуры с последующим уточнением найденных значений в полученном диапазоне методом Мюллера. Достоверность получаемых результатов подтверждена сравнением с известными численными решениями. В работе детально проанализированы зависимости низших частот колебаний от угла конусности и угла армирования свободно опертых, жестко закрепленных и консольных двухслойных и трехслойных композитных конических оболочек. Оценена возможность изменения низших частот и соответствующих им форм колебаний в зависимости от заданной комбинации угла конусности, условий закрепления краев конструкции, схемы укладки и угла армирования композиционного материала. Проведенная обширная серия вычислений позволила выявить наличие конфигураций, для которых низшие частоты превышают значения, соответствующие эквивалентной слоистой круговой цилиндрической оболочке.

Ключевые слова

классическая теория оболочекпрямая коническая оболочкаслоистый материалметод ортогональной прогонки Годуноваидеальная сжимаемая жидкостьметод обобщенных дифференциальных квадратурсобственные колебания.

Bibliography

1.Lindholm U.S., Hu W.C.L. Non-symmetric transverse vibrations of truncated conical shells, Int. J. Mech. Sci., 1966, vol. 8, pp. 561-579. doi: 10.1016/0020-7403(66)90078-6 Scholar
2.Shu C. An efficient approach for free vibration analysis of conical shells, Int. J. Mech. Sci., 1996, vol. 38, no. 8-9, pp. 935-949. doi: 10.1016/0020-7403(95)00096-8 Scholar
3.Qatu M.S. Vibration of laminated shells and plates, Oxford, Academic Press, 2004, 426 p. Scholar
4.Yan K., Zhang Y., Cai H., Tahouneh V. Vibrational characteristic of FG porous conical shells using Donnell's shell theory, Steel Compos. Struct., 2020, vol. 35, no. 2, pp. 249-260. doi: 10.12989/scs.2020.35.2.249 Scholar
5.Vescovini R., Fantuzzi N. Free vibrations of conical shells via Ritz method, Int. J. Mech. Sci., 2023, vol. 241, 107925. Scholar
6.Sheinman I., Greif S. Dynamic analysis of laminated shells of revolution, J. Compos. Mater., 1984, vol. 18, no. 3, pp. 200-215. doi: 10.1177/002199838401800301 Scholar
7.Sheinman I., Weissman S. Coupling between symmetric and antisymmetric modes in shells of revolution, J. Compos. Mater., 1987, vol. 21, no. 11, pp. 988-1007. doi: 10.1177/002199838702101101 Scholar
8.Sivadas K.R., Ganesan N. Effect of coupling between symmetric and antisymmetric modes in composite thick shells of revolution, Finite Elem. Anal. Des., 1992, vol. 11, no. 1, pp. 9-18. doi: 10.1016/0168-874X(92)90025-8 Scholar
9.Xi Z.C., Yam L.H., Leung T.P. Semi-analytical study of free vibration of composite shells of revolution based on the Reissner-Mindlin assumption, Int. J. Solids Struct., 1996, vol. 33, no. 6, pp. 851-863. doi: 10.1016/0020-7683(95)00063-G Scholar
10.Kayran A., Vinson J.R. Free vibration analysis of laminated composite truncated circular conical shells, AIAA J., 1990, vol. 28, no. 7, pp. 1259-1269. doi: 10.2514/3.25203 Scholar
11.Sivadas K.R., Ganesan N. Vibration analysis of laminated conical shells with variable thickness, J. Sound Vib., 1991, vol. 148, pp. 477-491. doi: 10.1016/0022-460X(91)90479-4 Scholar
12.Tong L. Free vibration of composite laminated conical shells, Int. J. Mech. Sci., 1993, vol. 35, no. 1, pp. 47-61. doi: 10.1016/0020-7403(93)90064-2 Scholar
13.Tong L. Free vibration of laminated conical shells including transverse shear deformation, Int. J. Solids Struct., 1994, vol. 31, no. 4, pp. 443-456. doi: 10.1016/0020-7683(94)90085-X Scholar
14.Andreyev A.N. Free oscillations of layered elastic composite shells of revolution, J. Appl. Mech. Tech. Phys., 1995, vol. 36, pp. 764-772. doi: 10.1007/BF02369291 Scholar
15.Shu C. Free vibration analysis of composite laminated conical shells by generalized differential quadrature, J. Sound Vib., 1996, vol. 194, no. 4, pp. 587-604. doi: 10.1006/jsvi.1996.0379 Scholar
16.Wu C.-P., Lee C.-Y. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness, Int. J. Mech. Sci., 2001, vol. 43, no. 8, pp. 1853-1869. doi: 10.1016/S0020-7403(01)00010-8 Scholar
17.Mehditabar A., Rahimi G.H., Fard K.M. Vibrational responses of antisymmetric angle-ply laminated conical shell by the methods of polynomial based differential quadrature and Fourier expansion based differential quadrature, Appl. Math. Comput., 2018, vol. 320, pp. 580-595. doi: 10.1016/j.amc.2017.10.017 Scholar
18.Wu C.P., Wu C.H. Asymptotic differential quadrature solutions for the free vibration of laminated conical shells, Comput. Mech., 2000, vol. 25, pp. 346-357. doi: 10.1007/s004660050482 Scholar
19.Hu H.-T., Ou S.-C. Maximization of the fundamental frequencies of laminated truncated conical shells with respect to fiber orientations, Compos. Struct., 2001, vol. 52, no. 3-4, pp. 265-275. doi: 10.1016/S0263-8223(01)00019-8 Scholar
20.Hu H.-T., Chen P.-J. Maximization of fundamental frequencies of axially compressed laminated truncated conical shells against fiber orientation, Thin-Walled Struct., 2015, vol. 97, pp. 154-170. doi: 10.1016/j.tws.2015.09.004 Scholar
21.Qu Y., Long X., Wu S., Meng G. A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia, Compos. Struct., 2013, vol. 98, pp. 169-191. doi: 10.1016/j.compstruct.2012.11.001 Scholar
22.Wu S., Qu Y., Hua H. Free vibration of laminated orthotropic conical shell on Pasternak foundation by a domain decomposition method, J. Compos. Mater., 2015, vol. 49, no. 1, pp. 35-52. doi: 10.1177/0021998313514259 Scholar
23.Sofiyev A.H., Kuruoglu N. Natural frequency of laminated orthotropic shells with different boundary conditions and resting on the Pasternak type elastic foundation, Compos. Part B Eng., 2011, vol. 42, no. 6, 1562-1570. doi: 10.1016/j.compositesb.2011.04.015 Scholar
24.Civalek O. The determination of frequencies of laminated conical shells via the discrete singular convolution method, J. Mech. Mater. Struct., 2006, vol. 1, no. 1, pp. 163-182. doi: 10.2140/joms.2006.1.163 Scholar
25.Civalek O. Free vibration analysis of composite conical shells using the discrete singular convolution algorithm, Steel Compos. Struct., 2006, vol. 6, no. 4, pp. 353-366. doi: 10.12989/scs.2006.6.4.353 Scholar
26.Civalek O. Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach, J. Comput. Appl. Math., 2007, vol. 205, no. 1, pp. 251-271. doi: 10.1016/j.cam.2006.05.001 Scholar
27.Civalek O. Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory, Compos. Part B Eng., 2013, vol. 45, no. 1, pp. 1001-1009. doi: 10.1016/j.compositesb.2012.05.018 Scholar
28.Tripathi V., Singh B.N., Shukla K.K. Free vibration of laminated composite conical shells with random material properties, Compos. Struct., 2007, vol. 81, no. 1, pp. 96-104. doi: 10.1016/j.compstruct.2006.08.002 Scholar
29.Xie X., Jin G., Li W., Liu Z. A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures, Compos. Struct., 2014, vol. 111, pp. 20-30. doi: 10.1016/j.compstruct.2013.12.019 Scholar
30.Kim J., Kim K., Kim K., Hong K., Paek C. Free vibration analysis of cross-ply laminated conical shell, cylindrical shell, and annular plate with variable thickness using the Haar wavelet discretization method, Shock Vib., 2022, vol. 2022, 6399675. doi: 10.1155/2022/6399675 Scholar
31.Ansari R., Faghih Shojaei M., Rouhi H., Hosseinzadeh M. A novel variational numerical method for analyzing the free vibration of composite conical shells, Appl. Math. Model., 2015, vol. 39, no. 10-11, pp. 2849-2860. doi: 10.1016/j.apm.2014.11.012 Scholar
32.Jin G., Ye T., Su Z. Structural vibration: A uniform accurate solution for laminated beams, plates and shells with general boundary conditions, Beijing, Science Press, 2015, 324 p. Scholar
33.Jin G., Ye T., Jia X., Gao S. A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints, Compos. Struct., 2014, vol. 109, pp. 150-168. doi: 10.1016/j.compstruct.2013.10.052 Scholar
34.Kouchakzadeh M.A., Shakouri M. Free vibration analysis of joined cross-ply laminated conical shells, Int. J. Mech. Sci., 2014, vol. 78, pp. 118-125. doi: 10.1016/j.ijmecsci.2013.11.008 Scholar
35.Izadi M.H., Hosseini-Hashemi S., Korayem M.H. Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory, Arch. Appl. Mech., 2018, vol. 88, no. 12, pp. 2231-2246. doi: 10.1007/s00419-018-1446-y Scholar
36.Shakouri M., Kouchakzadeh M.A. Analytical solution for vibration of generally laminated conical and cylindrical shells, Int. J. Mech. Sci., 2017, vol. 131-132, pp. 414-425. doi: 10.1016/j.ijmecsci.2017.07.016 Scholar
37.Sofiyev A.H. Application of the first order shear deformation theory to the solution of free vibration problem for laminated conical shells, Compos. Struct., 2018, vol. 188, pp. 340-346. doi: 10.1016/j.compstruct.2018.01.016 Scholar
38.Nasihatgozar M., Khalili S.M.R. Vibration and buckling analysis of laminated sandwich conical shells using higher order shear deformation theory and differential quadrature method, J. Sandw. Struct. Mater., 2019, vol. 21, no. 4, pp. 1445-1480. doi: 10.1177/1099636217715806 Scholar
39.Viswanathan K.K., Navaneethakrishnan P.V. Free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines, Int. J. Solids Struct. 2005, vol. 42, no. 3-4, pp. 1129-1150. doi: 10.1016/j.ijsolstr.2004.06.065 Scholar
40.Viswanathan K.K., Aziz Z.A., Javed S., Yaacob Y., Pullepu B. Free vibration of symmetric angle ply truncated conical shells under different boundary conditions using spline method, J. Mech. Sci. Tech., 2015, vol. 29, pp. 2073-2080. doi: 10.1007/s12206-015-0428-z Scholar
41.Viswanathan K.K., Javed S., Aziz Z.A. Free vibration of symmetric angle-ply layered conical shell frusta of variable thickness under shear deformation theory, Struct. Eng. Mech. 2013, vol. 45, no. 2, pp. 259-275. doi: 10.12989/sem.2013.45.2.259 Scholar
42.Viswanathan K.K., Javed S., Prabakar K., Aziz Z.A., Bakar I.A. Free vibration of anti-symmetric angle-ply laminated conical shells, Compos. Struct., 2015, vol. 122, pp. 488-495. doi: 10.1016/j.compstruct.2014.11.075 Scholar
43.Viswanathan K.K., Lee J.H., Aziz Z.A., Hossain I., Rongqiao W., Abdullah H.Y. Vibration analysis of cross-ply laminated truncated conical shells using a spline method, J. Eng. Math., 2012, vol. 76, pp. 139-156. doi: 10.1007/s10665-011-9525-x Scholar
44.Viswanathan K.K., Nor Hafizah A.K., Aziz Z.A. Free vibration of angle-ply laminated conical shell frusta with linear and exponential thickness variations, Int. J. Acoust. Vib., 2018, vol. 23, no. 2, pp. 264-276. doi: 10.20855/ijav.2018.23.21164 Scholar
45.Javed S., Al Mukaha F.H.H., Salama M.A. Free vibration analysis of composite conical shells with variable thickness, Shock Vib., 2020, vol. 2020, 4028607. doi: 10.1155/2020/4028607 Scholar
46.Javed S., Viswanathan K.K., Aziz Z.A., Lee J.H. Vibration analysis of a shear deformed anti-symmetric angle-ply conical shells with varying sinusoidal thickness, Struct. Eng. Mech., 2016, vol. 58, no. 6, pp. 1001-1020. doi: 10.12989/sem.2016.58.6.1001 Scholar
47.Guo S., Hu P., Li S. Free vibration analysis of composite conical shells using Walsh series method, Mater. Res. Express, 2021, vol. 8, no. 7, 075303. doi: 10.1088/2053-1591/ac0eb7 Scholar
48.Hu S., Zhong R., Wang Q., Bin Q. Vibration analysis of closed laminate conical, cylindrical shells and annular plates using meshfree method, Eng. Anal. Bound. Elem., 2021, vol. 133, pp. 341-361. doi: 10.1016/j.enganabound.2021.09.011 Scholar
49.He D., Shi D., Wang Q., Ma C. A unified power series method for vibration analysis of composite laminate conical, cylindrical shell and annular plate, Structures, 2021, vol. 29, pp. 305-327. doi: 10.1016/j.istruc.2020.11.016 Scholar
50.Zhu H., Wu J. Free vibration of partially fluid-filled or fluid-surrounded composite shells using the dynamic stiffness method, Acta Mech., 2020, vol. 231, pp. 3961-3978. doi: 10.1007/s00707-020-02734-3 Scholar
51.Paknejad R., Ghasemi F.A., Malekzadeh Fard K. Natural frequency analysis of multilayer truncated conical shells containing quiescent fluid on elastic foundation with different boundary conditions, Int. J. Appl. Mech., 2021, vol. 13, no. 07, 2150075. doi: 10.1142/S1758825121500757 Scholar
52.Nurul Izyan M.D., Viswanathan K.K., Aziz Z.A., Lee J.H., Prabakar K. Free vibration of layered truncated conical shells filled with quiescent fluid using spline method, Compos. Struct., 2017, vol. 163, no. 1, pp. 385-398. doi: 10.1016/j.compstruct.2016.12.011 Scholar
53.Godunov S.K. Ordinary differential equations with constant coefficients, American Mathematical Society, Providence, 1997. Scholar
54.Shu C. Differential quadrature and its application in engineering, London, Springer-Verlag, 2000, 356 p. Scholar
55.Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations of truncated conical shells containing fluid, Mech. Solids, 2022, vol. 57, no. 8, pp. 1971-1986. doi: 10.3103/S0025654422080064 Scholar
56.Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations of composite cylindrical shells partially filled with fluid, Vestnik St. Petersb. Univ. Math., 2023, vol. 56, no. 4, pp. 435-445. doi: 10.21638/spbu01.2023.313 Scholar
57.Bochkarev S.A., Matveenko V.P. Free vibration analysis of a cylindrical shell of variable thickness partially filled with fluid, Proc. Steklov Inst. Math., 2023, vol. 321(Suppl 1), pp. S20-S32. doi: 10.1134/S0081543823030045 Scholar
58.Bochkarev S.A., Lekomtsev S.V. Analysis of natural vibrations of truncated conical shells partially filled with fluid, Int. J. Mech. Sys. Dyn., 2024, vol. 4, no. 2, pp. 1-12. doi: 10.1002/msd2.12105 Scholar
59.Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N. Statika i dinamika tonkostennyh obolochechnyh konstrukcij [The Statics and Dynamics of Thin-walled Shell Structures]. Moscow, Mashinostroyeniy, 1975, 376 p. Scholar
60.Alfutov N.A., Zinov'ev P.A., Popov B.G. Raschet mnogoslojnyh plastin i obolochek iz kompozicionnyh materialov [Analysis of Multilayer Plates and Shells of Composite Materials]. Moscow, Mashinostroyeniy, 1984, 264 p. Scholar
61.Averbukh A.Z., Veitsman R.I., Genkin M.D. Kolebaniya elementov konstrukcii v zhidkosti [Vibration of Structural Elements in Fluid]. Moscow, Nauka, 1987, 158 p. Scholar
62.Muller D.E. A method for solving algebraic equations using an automatic computer, Math. Tables Other Aids Comput., 1956, 10, pp. 208-215. doi: 10.1090/S0025-5718-1956-0083822-0 Scholar
63.Wu X. Improved Muller method and Bisection method with global and asymptotic superlinear convergence of both point and interval for solving nonlinear equations, Appl. Math. Comput., 2005, vol. 166, no. 2, pp. 299-311. doi: 10.1016/j.amc.2004.04.120 Scholar
64.Narita Y., Ohta Y., Yamada G., Kobayashi Y. Analytical method for vibration of angle-ply cylindrical shells having arbitrary edges, AIAA J., 1992, vol. 30, no. 3, pp. 790-796. doi: 10.2514/3.10986 Scholar
65.Bochkarev S.A., Lekomtsev S.V. Natural vibrations and hydroelastic stability of laminated composite circular cylindrical shells, Struct. Eng. Mech., 2022, vol. 81, no. 6, pp. 769-780. doi: 10.12989/sem.2022.81.6.769 Scholar
66.Thambiratnam D.P., Zhuge Y. Axisymmetric free vibration analysis of conical shells, Eng. Struct., 1993, vol. 15, no. 2, pp. 83-89. doi: 10.1016/0141-0296(93)90002-L Scholar
67.Rahmanian M., Firouz-Abadi R.D., Cigeroglu E. Free vibrations of moderately thick truncated conical shells filled with quiescent fluid, J. Fluids Struct., 2016, vol. 63, pp. 280-301. doi: 10.1016/j.jfluidstructs.2016.04.005 Scholar
68.Buchanan G.R., Wong F.T.-I. Frequencies and mode shapes for thick truncated hollow cones, Int. J. Mech. Sci., 2001, vol. 43, no. 12, pp. 2815-2832. doi: 10.1016/S0020-7403(01)00068-6 Scholar

1.Lindholm, U.S. Non-symmetric transverse vibrations of truncated conical shells / U.S. Lindholm, W.C. Hu // Int. J. Mech. Sci. – 1966. – Vol. 8, no. 9. – P. 561–579. DOI: 10.1016/0020-7403(66)90078-6 Scholar
2.Shu, C. An efficient approach for free vibration analysis of conical shells / C. Shu // Int. J. Mech. Sci. – 1996. – Vol. 38, no. 8-9. – P. 935–949. DOI: 10.1016/0020-7403(95)00096-8. Scholar
3.Qatu, M.S. Vibration of Laminated Shells and Plates / M.S. Qatu. – Oxford: Academic Press, 2004. – 426 p. Scholar
4.Vibrational characteristic of FG porous conical shells using Donnell’s shell theory / K. Yan, Y. Zhang, H. Cai, V. Tahouneh // Steel Compos. Struct. – 2020. – Vol. 35, no. 2. – P. 249–260. DOI: 10.12989/scs.2020.35.2.249 Scholar
5.Vescovini, R. Free vibrations of conical shells via Ritz method / R. Vescovini, N. Fantuzzi // Int. J. Mech. Sci. – 2023. – Vol. 241. – 107925. DOI: 10.1016/j.ijmecsci.2022.107925 Scholar
6.Sheinman, I. Dynamic analysis of laminated shells of revolution / I. Sheinman, S. Greif // J. Compos. Mater. – 1984. – Vol. 18, no. 3. – P. 200–215. DOI: 10.1177/002199838401800301 Scholar
7.Sheinman, I. Coupling between symmetric and antisymmetric modes in shells of revolution / I. Sheinman, S. Weissman // J. Compos. Mater. – 1987. – Vol. 21, no. 11. – P. 988–1007. DOI: 10.1177/002199838702101101 Scholar
8.Sivadas, K.R. Effect of coupling between symmetric and antisymmetric modes in composite thick shells of revolution / K.R. Sivadas, N. Ganesan // Finite Elem. Anal. Des. – 1992. – Vol. 11, no. 1. – P. 9–18. DOI: 10.1016/0168-874X(92)90025-8 Scholar
9.Xi, Z.C. Semi-analytical study of free vibration of composite shells of revolution based on the Reissner-Mindlin assumption / Z.C. Xi, L.H. Yam, T.P. Leung // Int. J. Solids Struct. – 1996. – Vol. 33, no. 6. – P. 851–863. DOI: 10.1016/0020-7683(95)00063-G Scholar
10.Kayran, A. Free vibration analysis of laminated composite truncated circular conical shells / A. Kayran, J.R. Vinson // AIAA J. – 1990. – Vol. 28, no. 7. – P. 1259–1269. DOI: 10.2514/3.25203 Scholar
11.Sivadas, K.R. Vibration analysis of laminated conical shells with variable thickness / K.R. Sivadas, N. Ganesan // J. Sound Vib. – 1991. – Vol. 148, no. 3. – P. 477–491. DOI: 10.1016/0022-460X(91)90479-4 Scholar
12.Tong, L. Free vibration of composite laminated conical shells / L. Tong // Int. J. Mech. Sci. – 1993. – Vol. 35, no. 1. – P. 47–61. DOI: 10.1016/0020-7403(93)90064-2 Scholar
13.Tong, L. Free vibration of laminated conical shells including transverse shear deformation / L. Tong // Int. J. Solids Struct. – 1994. – Vol. 31, no. 4. – P. 443–456. DOI: 10.1016/0020-7683(94)90085-X Scholar
14.Андреев, А.Н. Свободные колебания слоистых упругих композитных оболочек вращения / А.Н. Андреев // Прикл. мех. техн. физ. – 1995. – Т. 36, № 5. – С. 145–154. Scholar
15.Shu, C. Free vibration analysis of composite laminated conical shells by generalized differential quadrature / C. Shu // J. Sound Vib. – 1996. – Vol. 194, no. 4. – P. 587–604. DOI: 10.1006/jsvi.1996.0379 Scholar
16.Wu, C.P. Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness / C.P. Wu, C.Y. Lee // Int. J. Mech. Sci. – 2001. – Vol. 43, no. 8. – P. 1853–1869. DOI: 10.1016/S0020-7403(01)00010-8 Scholar
17.Mehditabar, A. Vibrational responses of antisymmetric angle-ply laminated conical shell by the methods of polynomial based differential quadrature and Fourier expansion based differential quadrature / A. Mehditabar, G.H. Rahimi, K.M. Fard // Appl. Math. Comput. – 2018. – Vol. 320. – P. 580–595. DOI: 10.1016/j.amc.2017.10.017 Scholar
18.Wu, C.P. Asymptotic differential quadrature solutions for the free vibration of laminated conical shells / C.P. Wu, C.H. Wu // Comput. Mech. – 2000. – Vol. 25, no. 4. – P. 346–357. DOI: 10.1007/s004660050482 Scholar
19.Hu, H.T. Maximization of the fundamental frequencies of laminated truncated conical shells with respect to fiber orientations / H.T. Hu, S.C. Ou // Compos. Struct. – 2001. – Vol. 52, no. 3-4. – P. 265–275. DOI: 10.1016/S0263-8223(01)00019-8 Scholar
20.Hu, H.T. Maximization of fundamental frequencies of axially compressed laminated truncated conical shells against fiber orientation / H.T. Hu, P.J. Chen // Thin-Walled Struct. – 2015. – Vol. 97. – P. 154–170. DOI: 10.1016/j.tws.2015.09.004 Scholar
21.A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia / Y. Qu, X. Long, S. Wu, G. Meng // Compos. Struct. – 2013. – Vol. 98. – P. 169–191. DOI: 10.1016/j.comstruct.2012.11.001 Scholar
22.Wu, S. Free vibration of laminated orthotropic conical shell on Pasternak foundation by a domain decomposition method / S. Wu, Y. Qu, H. Hua // J. Compos. Mater. – 2015. – Vol. 49, no. 1. – P. 35–52. DOI: 10.1177/0021998313514259 Scholar
23.Sofiyev, A.H. Natural frequency of laminated orthotropic shells with different boundary conditions and resting on the Pasternak type elastic foundation / A.H. Sofiyev, N. Kuruoglu // Compos. Part B Eng. – 2011. – Vol. 42, no. 6. – P. 1562–1570. DOI: 10.1016/j.compositesb.2011.04.015 Scholar
24.Civalek, O. The determination of frequencies of laminated conical shells via the discrete singular convolution method / O. Civalek // J. Mech. Mater. Struct. – 2006. – Vol. 1, no. 1. – P. 163–182. DOI: 10.2140/joms.2006.1.163 Scholar
25.Civalek, O. Free vibration analysis of composite conical shells using the discrete singular convolution algorithm / O. Civalek // Steel Compos. Struct. – 2006. – Vol. 6, no. 4. – P. 353–366. DOI: 10.12989/scs.2006.6.4.353 Scholar
26.Civalek, O. Numerical analysis of free vibrations of laminated composite conical and cylindrical shells: Discrete singular convolution (DSC) approach / O. Civalek // J. Comput. Appl. Math. – 2007. – Vol. 205, no. 1. – P. 251–271. DOI: 10.1016/j.cam.2006.05.001 Scholar
27.Civalek, O. Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory / O. Civalek // Compos. Part B Eng. – 2013. – Vol. 45, no. 1. – P. 1001–1009. DOI: 10.1016/j.compositesb.2012.05.018 Scholar
28.Tripathi, V. Free vibration of laminated composite conical shells with random material properties / V. Tripathi, B.N. Singh, K.K. Shukla // Compos. Struct. – 2007. – Vol. 81, no. 1. – P. 96–104. DOI: 10.1016/j.comstruct.2006.08.002 Scholar
29.A numerical solution for vibration analysis of composite laminated conical, cylindrical shell and annular plate structures / X. Xiang, J. Guoyong, L. Wanyou, L. Zhigang // Compos. Struct. – 2014. – Vol. 111, no. 1. – P. 20–30. DOI: 10.1016/j.comstruct.2013.12.019 Scholar
30.Free vibration analysis of cross-ply laminated conical shell, cylindrical shell, and annular plate with variable thickness using the Haar wavelet discretization method / J. Kim, K. Kim, K. Kim, K. Hong, C. Pack // Shock Vib. – 2022. – Vol. 2022. – 6399675. DOI: 10.1155/2022/6399675 Scholar
31.A novel variational numerical method for analyzing the free vibration of composite conical shells / R. Ansari, M.F. Shojaei, H. Rouhi, M. Hosseinzadeh // Appl. Math. Model. – 2015. – Vol. 39, no. 10-11. – P. 2849–2860. DOI: 10.1016/j.apm.2014.11.012 Scholar
32.Jin, G. Structural vibration: A uniform accurate solution for laminated beams, plates and shells with general boundary conditions / G. Jin, T. Ye, Z. Su. – Beijing: Science Press, 2015. – 324 p. Scholar
33.A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints / G. Jin, T. Ye, X. Jia, S. Gao // Compos. Struct. – 2014. – Vol. 109, no. 1. – P. 150–168. DOI: 10.1016/j.comstruct.2013.10.052 Scholar
34.Kouchakzadeh, M.A. Free vibration analysis of joined cross-ply laminated conical shells / M.A. Kouchakzadeh, M. Shakouri // Int. J. Mech. Sci. – 2014. – Vol. 78. – P. 118–125. DOI: 10.1016/j.ijmecsci.2013.11.008 Scholar
35.Izadi, M.H. Analytical and FEM solutions for free vibration of joined cross-ply laminated thick conical shells using shear deformation theory / M.H. Izadi, S. Hosseini-Hashemi, M.H. Korayem // Arch. Appl. Mech. – 2018. – Vol. 88, no. 12. – P. 2231–2246. DOI: 10.1007/s00419-018-1446-y Scholar
36.Shakouri, M. Analytical solution for vibration of generally laminated conical and cylindrical shells / M. Shakouri, M.A. Kouchakzadeh // Int. J. Mech. Sci. – 2017. – Vol. 131-132. – P. 414–425. DOI: 10.1016/j.ijmecsci.2017.07.016 Scholar
37.Sofiyev, A.H. Application of the first order shear deformation theory to the solution of free vibration problem for laminated conical shells / A.H. Sofiyev // Compos. Struct. – 2018. – Vol. 188. – P. 340–346. DOI: 10.1016/j.comstruct.2018.01.016 Scholar
38.Nasihatgozar, M. Vibration and buckling analysis of laminated sandwich conical shells using higher order shear deformation theory and differential quadrature method / M. Nasihatgozar, S.M. Khalili // J. Sandw. Struct. Mater. – 2019. – Vol. 21, no. 4. – P. 1445–1480. DOI: 10.1177/1099636217715806 Scholar
39.Viswanathan K.K., Navaneethakrishnan P.V. Free vibration of layered truncated conical shell frusta of differently varying thickness by the method of collocation with cubic and quintic splines, Int. J. Solids Struct. 2005, vol. 42, no. 3-4, pp. 1129-1150. doi: 10.1016/j.ijsolstr.2004.06.065 Scholar
40.Viswanathan K.K., Aziz Z.A., Javed S., Yaacob Y., Pullepu B. Free vibration of symmetric angle ply truncated conical shells under different boundary conditions using spline method, J. Mech. Sci. Tech., 2015, vol. 29, pp. 2073-2080. doi: 10.1007/s12206-015-0428-z Scholar
41.Viswanathan K.K., Javed S., Aziz Z.A. Free vibration of symmetric angle-ply layered conical shell frusta of variable thickness under shear deformation theory, Struct. Eng. Mech. 2013, vol. 45, no. 2, pp. 259-275. doi: 10.12989/sem.2013.45.2.259 Scholar
42.Viswanathan K.K., Javed S., Prabakar K., Aziz Z.A., Bakar I.A. Free vibration of anti-symmetric angle-ply laminated conical shells, Compos. Struct., 2015, vol. 122, pp. 488-495. doi: 10.1016/j.compstruct.2014.11.075 Scholar
43.Viswanathan K.K., Lee J.H., Aziz Z.A., Hossain I., Rongqiao W., Abdullah H.Y. Vibration analysis of cross-ply laminated truncated conical shells using a spline method, J. Eng. Math., 2012, vol. 76, pp. 139-156. doi: 10.1007/s10665-011-9525-x Scholar
44.Viswanathan K.K., Nor Hafizah A.K., Aziz Z.A. Free vibration of angle-ply laminated conical shell frusta with linear and exponential thickness variations, Int. J. Acoust. Vib., 2018, vol. 23, no. 2, pp. 264-276. doi: 10.20855/ijav.2018.23.21164 Scholar
45.Javed S., Al Mukaha F.H.H., Salama M.A. Free vibration analysis of composite conical shells with variable thickness, Shock Vib., 2020, vol. 2020, 4028607. doi: 10.1155/2020/4028607 Scholar
46.Javed S., Viswanathan K.K., Aziz Z.A., Lee J.H. Vibration analysis of a shear deformed anti-symmetric angle-ply conical shells with varying sinusoidal thickness, Struct. Eng. Mech., 2016, vol. 58, no. 6, pp. 1001-1020. doi: 10.12989/sem.2016.58.6.1001 Scholar
47.Guo S., Hu P., Li S. Free vibration analysis of composite conical shells using Walsh series method, Mater. Res. Express, 2021, vol. 8, no. 7, 075303. doi: 10.1088/2053-1591/ac0eb7 Scholar
48.Hu S., Zhong R., Wang Q., Bin Q. Vibration analysis of closed laminate conical, cylindrical shells and annular plates using meshfree method, Eng. Anal. Bound. Elem., 2021, vol. 133, pp. 341-361. doi: 10.1016/j.enganabound.2021.09.011 Scholar
49.He D., Shi D., Wang Q., Ma C. A unified power series method for vibration analysis of composite laminate conical, cylindrical shell and annular plate, Structures, 2021, vol. 29, pp. 305-327. doi: 10.1016/j.istruc.2020.11.016 Scholar
50.Zhu H., Wu J. Free vibration of partially fluid-filled or fluid-surrounded composite shells using the dynamic stiffness method, Acta Mech., 2020, vol. 231, pp. 3961-3978. doi: 10.1007/s00707-020-02734-3 Scholar
51.Paknejad R., Ghasemi F.A., Malekzadeh Fard K. Natural frequency analysis of multilayer truncated conical shells containing quiescent fluid on elastic foundation with different boundary conditions, Int. J. Appl. Mech., 2021, vol. 13, no. 07, 2150075. doi: 10.1142/S1758825121500757 Scholar
52.Nurul Izyan M.D., Viswanathan K.K., Aziz Z.A., Lee J.H., Prabakar K. Free vibration of layered truncated conical shells filled with quiescent fluid using spline method, Compos. Struct., 2017, vol. 163, no. 1, pp. 385-398. doi: 10.1016/j.compstruct.2016.12.011 Scholar
53.Godunov S.K. Ordinary differential equations with constant coefficients, American Mathematical Society, Providence, 1997. Scholar
54.Shu C. Differential quadrature and its application in engineering, London, Springer-Verlag, 2000, 356 p. Scholar
55.Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations of truncated conical shells containing fluid, Mech. Solids, 2022, vol. 57, no. 8, pp. 1971-1986. doi: 10.3103/S0025654422080064 Scholar
56.Bochkarev S.A., Lekomtsev S.V., Matveenko V.P. Natural vibrations of composite cylindrical shells partially filled with fluid, Vestnik St. Petersb. Univ. Math., 2023, vol. 56, no. 4, pp. 435-445. doi: 10.21638/spbu01.2023.313 Scholar
57.Bochkarev S.A., Matveenko V.P. Free vibration analysis of a cylindrical shell of variable thickness partially filled with fluid, Proc. Steklov Inst. Math., 2023, vol. 321(Suppl 1), pp. S20-S32. doi: 10.1134/S0081543823030045 Scholar
58.Bochkarev S.A., Lekomtsev S.V. Analysis of natural vibrations of truncated conical shells partially filled with fluid, Int. J. Mech. Sys. Dyn., 2024, vol. 4, no. 2, pp. 1-12. doi: 10.1002/msd2.12105 Scholar
59.Karmishin A.V., Lyaskovets V.A., Myachenkov V.I., Frolov A.N. Statika i dinamika tonkostennyh obolochechnyh konstrukcij [The Statics and Dynamics of Thin-walled Shell Structures]. Moscow, Mashinostroyeniy, 1975, 376 p. Scholar
60.Alfutov N.A., Zinov'ev P.A., Popov B.G. Raschet mnogoslojnyh plastin i obolochek iz kompozicionnyh materialov [Analysis of Multilayer Plates and Shells of Composite Materials]. Moscow, Mashinostroyeniy, 1984, 264 p. Scholar
61.Averbukh A.Z., Veitsman R.I., Genkin M.D. Kolebaniya elementov konstrukcii v zhidkosti [Vibration of Structural Elements in Fluid]. Moscow, Nauka, 1987, 158 p. Scholar
62.Muller D.E. A method for solving algebraic equations using an automatic computer, Math. Tables Other Aids Comput., 1956, 10, pp. 208-215. doi: 10.1090/S0025-5718-1956-0083822-0 Scholar
63.Wu X. Improved Muller method and Bisection method with global and asymptotic superlinear convergence of both point and interval for solving nonlinear equations, Appl. Math. Comput., 2005, vol. 166, no. 2, pp. 299-311. doi: 10.1016/j.amc.2004.04.120 Scholar
64.Narita Y., Ohta Y., Yamada G., Kobayashi Y. Analytical method for vibration of angle-ply cylindrical shells having arbitrary edges, AIAA J., 1992, vol. 30, no. 3, pp. 790-796. doi: 10.2514/3.10986 Scholar
65.Bochkarev S.A., Lekomtsev S.V. Natural vibrations and hydroelastic stability of laminated composite circular cylindrical shells, Struct. Eng. Mech., 2022, vol. 81, no. 6, pp. 769-780. doi: 10.12989/sem.2022.81.6.769 Scholar
66.Thambiratnam D.P., Zhuge Y. Axisymmetric free vibration analysis of conical shells, Eng. Struct., 1993, vol. 15, no. 2, pp. 83-89. doi: 10.1016/0141-0296(93)90002-L Scholar
67.Rahmanian M., Firouz-Abadi R.D., Cigeroglu E. Free vibrations of moderately thick truncated conical shells filled with quiescent fluid, J. Fluids Struct., 2016, vol. 63, pp. 280-301. doi: 10.1016/j.jfluidstructs.2016.04.005 Scholar
68.Buchanan G.R., Wong F.T.-I. Frequencies and mode shapes for thick truncated hollow cones, Int. J. Mech. Sci., 2001, vol. 43, no. 12, pp. 2815-2832. doi: 10.1016/S0020-7403(01)00068-6 Scholar

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Published: 23 April 2025
Issue date: 23 April 2025
Issue: No. 1 (2025)