Nonlinear Vibration of Multilayer Nonhomogeneous Moderately-Thick Cylindrical Shells on Three-Parameter Nonlinear Foundations

Document Type : Research Paper

Authors
Abdullah H. Sofiyev 1 , 2 , 3
Tarkan Vergül 4 , 5
Fethi Kadıoglu 4
1 Department of Mathematics, Istanbul Ticaret University, Beyoglu, 34445, Istanbul, Türkiye
2 Scientific Research Department, Azerbaijan University of Architecture and Construction, Baku, 1073, Azerbaijan
3 Scientific Research Centers Odlar Yurdu University, AZ1073/Baku, Azerbaijan
4 Department of Civil Engineering, Istanbul Technical University, 34467, Sariyer/Istanbul, Türkiye
5 Department of Civil Engineering, Istanbul Ticaret University, Beyoglu, 34445, Istanbul, Türkiye
10.22055/jacm.2025.48952.5620
Abstract
This study focuses on the analysis of the nonlinear vibration behavior of nonhomogenous isotropic (NHI) multilayered cylindrical shells resting on three-parameter nonlinear elastic foundations (Winkler-Pasternak type) within a theoretical framework based on Donnell-type shell theory, which considers von Kármán-type geometric nonlinearities and transverse shear deformation effects. The stress-displacement relations for each NHI layer are defined within the context of generalized Hooke's law, and based on this, the governing equations of the system are derived in the form of partial differential equations. These equations are reduced to second-order time-dependent nonlinear ordinary differential equations using the Galerkin approximation. The nonlinear frequency-amplitude relationship is established using the semi-inverse method, which establishes a variational formulation through an assumed functional representation and provides an approximate analytical solution to the nonlinear governing equations. Furthermore, the study comprehensively examined the effects of nonlinear elastic foundation parameters (Winkler and Pasternak coefficients), shear deformations, degree of nonhomogeneity, number and arrangement of layers, and shell geometric ratios on nonlinear frequency characteristics, and new findings are presented that contribute to the literature.
Keywords
Multilayer moderately-thick cylindrical shells
Nonhomegeneity
three-parameter nonlinear foundations
Layer arrangement
Nonlinear vibration
Nonlinear frequencies
Subjects

Publisher’s Note Shahid Chamran University of Ahvaz remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

[1] Lomakin, V.A., Theory of Elasticity of Inhomogeneous Bodies, Moscow University, Moscow, 1976.
[2] Grigorenko, Y.M., Vasilenko, A.T., Pankratova, N.D., Elasticity Theory Problems for Heterogeneous Bodies, Naukova Dumka, Kiev, 1991  [in Russian].
[3] Sofiyev, A.H., Schnack, E., The buckling of cross-ply laminated non-homogeneous orthotropic composite conical thin shells under a dynamic external pressure, Acta Mechanica, 162(1), 2003, 29-40.
[4] Awrejcewicz, J., Krysko, V.A., Theory of Non-homogeneous Shells. In: Chaos in Structural Mechanics, Springer, Berlin, Heidelberg, 2008.
[5] Shen, H.S., Functionally Graded Materials: Nonlinear Analysis of Plates and Shells, CRC Press, Boca Raton, FL, 2009.
[6] Vol'mir, A.S., Nonlinear Dynamics of Plates and Shells, Nauka, Moscow, 1972 [in Russian].
[7] Leissa, A.W., Vibration of Shells, Scientific and Technical Information Office, National Aeronautics and Space Administration: Washington, DC, USA, 1973.
[8] Amabili, M., Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, 2008.
[9] Ambartsumyan, S.A., General Theory of Anisotropic Shells, Nauka, Moscow, 1974.
[10] Reddy, J.N., Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, New-York, 2004.
[11] Qatu, M.S., Vibration of Laminated Shells and Plates, Academic Press, 2004.
[12] Pasternak, P.L, On a New Method of an Elastic Foundation by Means of Two Foundation Constants, Gosstroyizdat, Moscow, 1954.
[13] Gorbunov-Possadov, M.I., Malikova, T.A., Solomin, V.I., Design of Structures on Elastic Foundation, Gos. Izd. Lit. Stroit. Arkh, Moscow, Russia, 1984.
[14] Younesian, D., Hosseinkhani, A., Askari, H., Esmailzadeh, E., Elastic and viscoelastic foundations: a review on linear and nonlinear vibration modeling and applications, Nonlinear Dynamics, 97(1), 2019, 853–895. 
[15] Shen, H.S., Postbuckling analysis of orthotropic rectangular plates on nonlinear elastic foundations, Engineering Structures, 17(6), 1995, 407-412.
[16] Luo, Y.F., Teng, J.G., Stability analysis of shells of revolution on nonlinear elastic foundations, Computers & Structures, 69(4), 1998, 499-511.
[17] Qian, Q., Wang, L., Ni, Q., Nonlinear responses of a fluid-conveying pipe embedded in nonlinear elastic foundations, Acta Mechanica Solida Sinica, 21(2), 2008, 170-176.
[18] Sofiyev, A.H., Karaca, Z., Zerin, Z., Non-linear vibration of composite orthotropic cylindrical shells on the non-linear elastic foundations within the shear deformation theory, Composite Structures, 159, 2017, 53–62.
[19] Ahmadi, H., Nonlinear primary resonance of imperfect spiral stiffened functionally graded cylindrical shells surrounded by damping and nonlinear elastic foundation, Engineering with Computers, 35(4), 2019, 1491-1505.
[20] Babaei, H., Kiani, Y., Eslami, M.R., Large amplitude free vibrations of long FGM cylindrical panels on nonlinear elastic foundation based on physical neutral surface, Composite Structures, 220, 2019, 888-898.
[21] Zhu, B., Xu, Q., Li, M., Li, YH., Nonlinear free and forced vibrations of porous functionally graded pipes conveying fluid and resting on nonlinear elastic foundation, Composite Structures, 252, 2020, 112672.
[22] Babaei, H., Eslami, M.R., On nonlinear vibration and snap-through buckling of long FG porous cylindrical panels using nonlocal strain gradient theory, Composite Structures, 256, 2021, 113125.
[23] Babaei, H., Eslami, M.R., Study on nonlinear vibrations of temperature- and size-dependent FG porous arches on elastic foundation using nonlocal strain gradient theory, The European Physical Journal Plus, 136(1), 2021, 24, 
[24] Babaei, H., Thermomechanical analysis of snap-buckling phenomenon in long FG-CNTRC cylindrical panels resting on nonlinear elastic foundation, Composite Structures, 286, 2022, 115199.
[25] Zemlyanukhin, A.I., Bochkarev, A.V., Erofeev, V.I., Ratushny, A.V., Axisymmetric longitudinal waves in a cylindrical shell interacting with a nonlinear elastic medium, Wave Motion, 114, 2022, 103020.
[26] Wang, S.X., Zhang, Y.P., Jiang, G., Wen, M., Wu, W.B., Simplified approach for analyzing thermal buckling of pipeline based on nonlinear Pasternak foundation model, Ocean Engineering, 314, 2024, 119731. 
[27] Hoang, V.N.V., Thanh, P.T., Toledo, L., Impact of non-homogeneous Winkler-Pasternak foundation on nonlinear dynamic characteristics of fluid-conveying functionally graded cylindrical shells, Ocean Engineering, 307, 2024, 118123.
[28] Dai, L.M., Foroutan, K., Sub-harmonic and super-harmonic resonances of spiral stiffened FG cylindrical panels of variable thickness embedded within a nonlinear elastic foundation, Journal of Vibration and Control, 31(3-4), 2025, 412-430.
[29] Shen, H.S., Williams, F.W., Postbuckling analysis of imperfect composite laminated plates on non-linear elastic foundations, International Journal of Non-Linear Mechanics, 30(5), 1995, 651-659.
[30] Tornabene, F., Reddy, J.N., FGM and laminated doubly-curved and degenerate shells resting on nonlinear elastic foundations: a GDQ solution for static analysis with a posteriori stress and strain recovery, Journal of the Indian Institute of Science, 93(4), 2013, 635-688.
[31] Tornabene, F., Fantuzzi, N., Bacciocchi, M., Reddy, J.N., A posteriori stress and strain recovery procedure for the static analysis of laminated shells resting on nonlinear elastic foundation, Composites Part B: Engineering, 126, 2017, 162-191.
[32] Fard, K.M., Bohlooly, M., Postbuckling of piezolarninated cylindrical shells with eccentrically/concentrically stiffeners surrounded by nonlinear elastic foundations, Composite Structures, 171, 2017, 360-369.
[33] Foroutan, K., Varedi-Koulaei, S.M., Duc, N.D., Ahmadi, H., Non-linear static and dynamic buckling analysis of laminated composite cylindrical shell embedded in non-linear elastic foundation using the swarm-based metaheuristic algorithms, European Journal of Mechanics - A/Solids, 91, 2022, 104420.
[34] Zhang, P., Meng, Z., Wei, H., Liu, N., An analytical investigation on the nonlinear vibration behavior of a new hybrid laminated nanocomposite cylindrical shell resting on the three-parameter nonlinear substrate, Journal of Vibration Engineering & Technologies, 12, 2024, 77-96.
[35] Herisanu, N., Marinca, B., Marinca, V., On the nonlinear forced vibration of the magnetostrictive laminated beam in a complex environment, Mathematics, 12(23), 2024, 3836.
[36] Chen, H.Y., Li, W., Shi, HT., Coupled response of sandwich beams on the nonlinear foundation under Galerkin truncation convergence criterion, International Journal of Dynamics and Control, 12(7), 2024, 2144-2154.
[37] Alimoradzadeh, M., Akbas, S.D., The influence of temperature on nonlinear vibration of laminated composite beams covered by FGM layers embedded on nonlinear viscoelastic foundation subjected to moving load, Structures, 73, 2025, 108352.
[38] Sobhy, M., A comprehensive nonlinear analysis of asymmetric nanocomposite sandwich beams with a negative Poisson's ratio core on nonlinear elastic foundation under hygrothermal conditions, ZAMM ‐ Journal of Applied Mathematics and Mechanics, 105(5), 2025, e70045.
[39] Sofiyev, A.H., Avey, M., Influences of nonlinear elastic foundations on the nonlinear vibration of shear deformable laminated cylindrical shells, ZAMM - Journal of Applied Mathematics and Mechanics, 105(9), 2025, e70213.
[40] He, J.H., Variational principles for some nonlinear partial differential equations with variable coefficients, Chaos, Solitons & Fractals, 19(4), 2004, 847-851.
[41] Wang, Q., Shi, D., Shi, X., A modified solution for the free vibration analysis of moderately thick orthotropic rectangular plates with general boundary conditions, internal line supports and resting on elastic foundation, Meccanica, 51, 2016, 1985–2017.
[42] Zhou, D., Cheung, Y.K., Lo, S.H., Au, F.T.K., Three-dimensional vibration analysis of rectangular thick plates on Pasternak foundation, International Journal for Numerical Methods in Engineering, 59, 2004, 1313-1334.
Journal of Applied and Computational Mechanics
Volume 12, Issue 3
Summer 2026
Pages 1149-1163