Free and Forced Vibration Analysis of Carbon/Glass Hybrid Composite Laminated Plates Under Arbitrary Boundary Conditions
Source abstract
Abstract
Abstract This paper presents the theoretical investigations on the free and forced vibration behaviours of carbon/glass hybrid composite laminated plates with arbitrary boundary conditions. The unknown allowable displacement functions of the physical middle surface are expressed in terms of standard cosine Fourier series and sinusoidal auxiliary functions to ensure the continuity of the displacement functions and their derivatives at the structural boundaries. Arbitrary boundary conditions are achieved through the introduction of an artificial spring technique. The first shear deformation theory and Lagrange equations are utilized to derive the energy expression, and the eigenvalue equations associated with free and forced vibration are obtained by Rayleigh-Ritz variational operations. Subsequently, these equations are then solved to determine the natural frequency, mode of vibration, and the steady-state displacement response under forced excitation. The new results are compared with those from references and finite element methods to verify the convergence, accuracy and efficiency of the analytical method. The effects of hybrid ratios, stacking sequences, lamination schemes, fibre orientation, boundary conditions and excitation force on the free and forced vibration behaviours of the carbon/glass hybrid composite laminated plates are analyzed in detail.
Source: crossref · Quality: baseline
Research context
OpenAlex topics: Composite Structure Analysis and Optimization · Structural Analysis of Composite Materials · Mechanical Behavior of Composites
SciVX discovery tags are terminology-based. OpenAlex topics are source-provided classifications.
References (74)
Show source-provided references
1. Belbachir, N., Bourada, F., Bousahla, A.A., Tounsi, A., Al-Osta, M.A., Ghazwani, M.H., Alnujaie, A., Tounsi, A.: A refined quasi-3D theory for stability and dynamic investigation of cross-ply laminated composite plates on Winkler-Pasternak foundation. Struct. Eng. Mech. 85, 433 (2023)
2. Zhang, X., Dai, W., Cai, B., Li, C., Huang, W., Fang, C.: Numerical and experimental investigation of bearing capacity for compressed stiffened composite panel with different stringer section geometries. Appl. Compos. Mater. 29, 1507–1535 (2022). https://doi.org/10.1007/s10443-022-10030-710.1007/s10443-022-10030-7 ↗
3. Chen, N.-Z., Guedes Soares, C.: Longitudinal strength analysis of ship hulls of composite materials under sagging moments. Compos. Struct. 77, 36–44 (2007). https://doi.org/10.1016/j.compstruct.2005.06.00210.1016/j.compstruct.2005.06.002 ↗
4. Chen, N.-Z., Guedes Soares, C.: Reliability assessment for ultimate longitudinal strength of ship hulls in composite materials. Probabilistic Eng. Mech. 22, 330–342 (2007). https://doi.org/10.1016/j.probengmech.2007.05.00110.1016/j.probengmech.2007.05.001 ↗
5. Alizadeh, F., Sebdani, M., Guedes Soares, C.: Numerical analysis of the residual ultimate strength of composite laminates under uniaxial compressive load. Compos Struct 300, 116161 (2022). https://doi.org/10.1016/j.compstruct.2022.11616110.1016/j.compstruct.2022.116161 ↗
6. Li, M., Yan, R., Xu, L., Guedes Soares, C.: A general framework of higher-order shear deformation theories with a novel unified plate model for composite laminated and FGM plates. Compos. Struct. 261, 113560 (2021). https://doi.org/10.1016/j.compstruct.2021.11356010.1016/j.compstruct.2021.113560 ↗
7. Li, M., Yan, R., Shen, W., Qin, K., Li, J., Liu, K.: Fatigue characteristics of sandwich composite joints in ships. Ocean Eng. 254, 111254 (2022). https://doi.org/10.1016/j.oceaneng.2022.11125410.1016/j.oceaneng.2022.111254 ↗
8. Li, M., Liu, Z., Yan, R., Lu, J., Guedes Soares, C.: Experimental and numerical investigation on composite single-lap single-bolt sandwich joints with different geometric parameters. Mar. Struct. 85, 103259 (2022). https://doi.org/10.1016/j.marstruc.2022.10325910.1016/j.marstruc.2022.103259 ↗
9. Alizadeh, F., Guedes Soares, C.: Experimental and numerical investigation of the fracture toughness of Glass/Vinylester composite laminates. Eur. J. Mech. A Solids 73, 204–211 (2019). https://doi.org/10.1016/j.euromechsol.2018.08.00310.1016/j.euromechsol.2018.08.003 ↗
10. Kharghani, N., Guedes Soares, C.: Experimental, numerical and analytical study of bending of rectangular composite laminates. Eur. J. Mech. A Solids 72, 155–174 (2018). https://doi.org/10.1016/j.euromechsol.2018.05.00710.1016/j.euromechsol.2018.05.007 ↗
11. Mantari, J.L., Bonilla, E.M., Guedes Soares, C.: A new tangential-exponential higher order shear deformation theory for advanced composite plates. Compos. Part B Eng. 60, 319–328 (2014). https://doi.org/10.1016/j.compositesb.2013.12.00110.1016/j.compositesb.2013.12.001 ↗
12. Barsotti, B., Gaiotti, M., Rizzo, C.M.: Recent industrial developments of marine composites limit states and design approaches on strength. J. Mar. Sci. Appl. 19, 553–566 (2020). https://doi.org/10.1007/s11804-020-00171-110.1007/s11804-020-00171-1 ↗
13. Zuo, P., Srinivasan, D.V., Vassilopoulos, A.P.: Review of hybrid composites fatigue. Compos. Struct. 274, 114358 (2021). https://doi.org/10.1016/j.compstruct.2021.11435810.1016/j.compstruct.2021.114358 ↗
14. Van Vinh, P., Tounsi, A.: Free vibration analysis of functionally graded doubly curved nanoshells using nonlocal first-order shear deformation theory with variable nonlocal parameters. Thin-Walled Struct. 174, 109084 (2022)10.1016/j.tws.2022.109084 ↗
15. Naito, K.: Flexural properties of carbon/glass hybrid thermoplastic epoxy composite rods under static and fatigue loadings. Appl. Compos. Mater. 28, 753–766 (2021). https://doi.org/10.1007/s10443-021-09893-z10.1007/s10443-021-09893-z ↗
16. Li, M., Guedes Soares, C., Yan, C.: Free vibration analysis of FGM plates on Winkler/Pasternak/Kerr foundation by using a simple quasi-3D HSDT. Compos Struct 264, 113643 (2021). https://doi.org/10.1016/j.compstruct.2021.11364310.1016/j.compstruct.2021.113643 ↗
17. Diniz, C.A., Pereira, J.L.J., da Cunha, S.S., Gomes, G.F.: Drop-off location optimization in hybrid CFRP/GFRP composite tubes using design of experiments and SunFlower optimization algorithm. Appl. Compos. Mater. 29, 1841–1870 (2022). https://doi.org/10.1007/s10443-022-10046-z10.1007/s10443-022-10046-z ↗
18. Monticeli, F.M., Cioffi, M.O.H., Voorwald, H.J.C.: Mode II delamination of carbon-glass fiber/epoxy hybrid composite under fatigue loading. Int. J. Fatigue 154, 106574 (2022). https://doi.org/10.1016/j.ijfatigue.2021.10657410.1016/j.ijfatigue.2021.106574 ↗
19. Hu, B., Li, Y., Jiang, Y.-T., Tang, H.-Z.: Bond behavior of hybrid FRP-to-steel joints. Compos. Struct. 237, 111936 (2020). https://doi.org/10.1016/j.compstruct.2020.11193610.1016/j.compstruct.2020.111936 ↗
20. Sajid, Z., Karuppanan, S., Kee, K.E., Sallih, N., Shah, S.Z.H.: Carbon/basalt hybrid composite bolted joint for improved bearing performance and cost efficiency. Compos. Struct. 275, 114427 (2021). https://doi.org/10.1016/j.compstruct.2021.11442710.1016/j.compstruct.2021.114427 ↗
21. Damghani, M., Saddler, J., Sammon, E., Atkinson, G.A., Matthews, J., Murphy, A.: An experimental investigation of the impact response and post-impact shear buckling behaviour of hybrid composite laminates. Compos. Struct. 305, 116506 (2023). https://doi.org/10.1016/j.compstruct.2022.11650610.1016/j.compstruct.2022.116506 ↗
22. Vummadisetti, S., Singh, S.B.: Buckling and postbuckling response of hybrid composite plates under uniaxial compressive loading. J. Build. Eng. 27, 101002 (2020). https://doi.org/10.1016/j.jobe.2019.10100210.1016/j.jobe.2019.101002 ↗
23. Li, M., Zhang, P., Cui, J., Liu, Z., Zhao, Y., Qiu, Y., Li, H.: Effect of hybridization on free vibration response of Carbon/Glass hybrid composite laminates. J. Reinf. Plast. Compos. (2023). https://doi.org/10.1177/0731684423119729810.1177/07316844231197298 ↗
24. Lowde, M.J., Peters, H.G., Geraghty, R., Graham-Jones, J., Pemberton, R., Summerscales, J.: The 100 m composite ship? J. Mar. Sci. Eng. (2022). https://doi.org/10.3390/jmse1003040810.3390/jmse10030408 ↗
25. Jiang, H., Liu, X., Jiang, S., Ren, Y.: Hybrid effects and interactive failure mechanisms of hybrid fiber composites under flexural loading: Carbon/Kevlar, carbon/glass, carbon/glass/Kevlar. Aerosp. Sci. Technol. 133, 108105 (2023). https://doi.org/10.1016/j.ast.2023.10810510.1016/j.ast.2023.108105 ↗
26. Yang, G., Guo, H., Xiao, H., Jiang, H., Liu, R.: Out-of-plane stiffness analysis of kevlar/carbon fiber hybrid composite skins for a shear variable-sweep wing. Appl. Compos. Mater. 28, 1653–1673 (2021). https://doi.org/10.1007/s10443-021-09926-710.1007/s10443-021-09926-7 ↗
27. Chen, D., Sun, G., Meng, M., Jin, X., Li, Q.: Flexural performance and cost efficiency of carbon/basalt/glass hybrid FRP composite laminates. Thin-Walled Struct. 142, 516–531 (2019). https://doi.org/10.1016/j.tws.2019.03.05610.1016/j.tws.2019.03.056 ↗
28. Wang, L., Ma, W., Deng, L., Liu, S., Yang, T.: Mechanical model and mechanical property analysis of fibre-reinforced hybrid composite pipes. Mar. Struct. 89, 103396 (2023). https://doi.org/10.1016/j.marstruc.2023.10339610.1016/j.marstruc.2023.103396 ↗
29. Sun, G., Tong, S., Chen, D., Gong, Z., Li, Q.: Mechanical properties of hybrid composites reinforced by carbon and basalt fibers. Int. J. Mech. Sci. 148, 636–651 (2018). https://doi.org/10.1016/j.ijmecsci.2018.08.00710.1016/j.ijmecsci.2018.08.007 ↗
30. Yao, Y., Cui, J., Wang, S., Xu, L., Li, G., Pan, H., Bai, X.: Comparison of tensile properties of carbon fiber, basalt fiber and hybrid fiber reinforced composites under various strain rates. Appl. Compos. Mater. 29, 1147–1165 (2022). https://doi.org/10.1007/s10443-022-10012-910.1007/s10443-022-10012-9 ↗
31. Yang, G., Guo, H., Xiao, H., Liu, R., Shi, C.: Shear-driving force and critical shear angle analysis of kevlar/carbon fiber hybrid composite skins for a shear variable-sweep wing based on the classical plate theory. Appl. Compos. Mater. 29, 1871–1887 (2022). https://doi.org/10.1007/s10443-022-10044-110.1007/s10443-022-10044-1 ↗
32. Wang, Z., Luo, Q., Li, Q., Sun, G.: Design optimization of bioinspired helicoidal CFRPP/GFRPP hybrid composites for multiple low-velocity impact loads. Int. J. Mech. Sci. 219, 107064 (2022). https://doi.org/10.1016/j.ijmecsci.2022.10706410.1016/j.ijmecsci.2022.107064 ↗
33. da Silva, A.A.X., Scazzosi, R., Manes, A., Amico, S.C.: High-velocity impact behavior of Aramid/S2-glass interply hybrid laminates. Appl. Compos. Mater. 28, 1899–1917 (2021). https://doi.org/10.1007/s10443-021-09946-310.1007/s10443-021-09946-3 ↗
34. Bian, T., Lyu, Q., Fan, X., Zhang, X., Li, X., Guo, Z.: Effects of fiber architectures on the impact resistance of composite laminates under low-velocity impact. Appl. Compos. Mater. 29, 1125–1145 (2022). https://doi.org/10.1007/s10443-022-10009-410.1007/s10443-022-10009-4 ↗
35. Zafar, H.M.N., Nair, F.: Comparison of static/dynamic loading and tensile behavior of interply and intraply hybridized carbon/basalt epoxy composites. Appl. Compos. Mater. 29, 451–472 (2022). https://doi.org/10.1007/s10443-021-09973-010.1007/s10443-021-09973-0 ↗
36. Masoumi, A., Shojaeefard, M.H., Najibi, A.: Comparison of steel, aluminum and composite bonnet in terms of pedestrian head impact. Saf. Sci. 49, 1371–1380 (2011). https://doi.org/10.1016/j.ssci.2011.05.00810.1016/j.ssci.2011.05.008 ↗
37. Bhattacharjee, A., Ganguly, K., Roy, H.: An operator based novel micromechanical model of viscoelastic hybrid woven fibre-particulate reinforced polymer composites. Eur. J. Mech. A Solids 83, 104044 (2020). https://doi.org/10.1016/j.euromechsol.2020.10404410.1016/j.euromechsol.2020.104044 ↗
38. Chen, C.-S., Chen, W.-R., Chien, R.-D.: Stability of parametric vibrations of hybrid composite plates. Eur. J. Mech. A Solids 28, 329–337 (2009). https://doi.org/10.1016/j.euromechsol.2008.06.00410.1016/j.euromechsol.2008.06.004 ↗
39. Roy, S., Thakur, S.N., Ray, C.: Free vibration analysis of laminated composite hybrid and GFRP shells based on higher order zigzag theory with experimental validation. Eur. J. Mech. A Solids 88, 104261 (2021). https://doi.org/10.1016/j.euromechsol.2021.10426110.1016/j.euromechsol.2021.104261 ↗
40. Sahu, P., Sharma, N., Panda, S.K.: Numerical prediction and experimental validation of free vibration responses of hybrid composite (Glass/Carbon/Kevlar) curved panel structure. Compos. Struct. 241, 112073 (2020). https://doi.org/10.1016/j.compstruct.2020.11207310.1016/j.compstruct.2020.112073 ↗
41. Murugan, R., Ramesh, R., Padmanabhan, K.: Investigation of the mechanical behavior and vibration characteristics of thin walled glass/carbon hybrid composite beams under a fixed-free boundary condition. Mech. Adv. Mater. Struct. 23, 909–916 (2016). https://doi.org/10.1080/15376494.2015.105639410.1080/15376494.2015.1056394 ↗
42. Kumar, K.S., Siva, I., Rajini, N., Jeyaraj, P., Jappes, J.W.: Tensile, impact, and vibration properties of coconut sheath/sisal hybrid composites: effect of stacking sequence. J. Reinf. Plast. Compos. 33, 1802–1812 (2014). https://doi.org/10.1177/073168441454678210.1177/0731684414546782 ↗
43. Nega, B.F., Pierce, R.S., Yi, X., Liu, X.: Characterization of mechanical and damping properties of carbon/jute fibre hybrid SMC composites. Appl. Compos. Mater. 29, 1637–1651 (2022). https://doi.org/10.1007/s10443-022-10034-310.1007/s10443-022-10034-3 ↗
44. Bulut, M., Erkliğ, A., Yeter, E.: Experimental investigation on influence of Kevlar fiber hybridization on tensile and damping response of Kevlar/glass/epoxy resin composite laminates. J. Compos. Mater. 50, 1875–1886 (2016). https://doi.org/10.1177/002199831559755210.1177/0021998315597552 ↗
45. Barai, A., Durvasula, S.: Vibration and buckling of hybrid laminated curved panels. Compos. Struct. 21, 15–27 (1992). https://doi.org/10.1016/0263-8223(92)90076-O10.1016/0263-8223(92)90076-O ↗
46. Bhudolia, S.K., Kam, K.K., Joshi, S.C.: Mechanical and vibration response of insulated hybrid composites. J. Ind. Text. 47, 1887–1907 (2018). https://doi.org/10.1177/152808371771448110.1177/1528083717714481 ↗
47. Tiwari, S., Hirwani, C.K., Barman, A.G.: Eigenfrequency behavior of banana/glass/epoxy hybrid composite-a numerical and experimental investigation. J. Nat. Fibers 19, 9514–9530 (2022). https://doi.org/10.1080/15440478.2021.198283810.1080/15440478.2021.1982838 ↗
48. Bulut, M., Bozkurt, Ö.Y., Erkliğ, A.: Damping and vibration characteristics of basalt-aramid/epoxy hybrid composite laminates. J. Polym. Eng. 36, 173–180 (2016). https://doi.org/10.1515/polyeng-2015-016810.1515/polyeng-2015-0168 ↗
49. Assarar, M., Zouari, W., Sabhi, H., Ayad, R., Berthelot, J.-M.: Evaluation of the damping of hybrid carbon–flax reinforced composites. Compos. Struct. 132, 148–154 (2015). https://doi.org/10.1016/j.compstruct.2015.05.01610.1016/j.compstruct.2015.05.016 ↗
50. Lee, Y.-S., Kim, Y.-W.: Analysis of nonlinear vibration of hybrid composite plates. Comput. Struct. 61, 573–578 (1996). https://doi.org/10.1016/0045-7949(96)00055-710.1016/0045-7949(96)00055-7 ↗
51. Pingulkar, H., Mache, A., Munde, Y., Siva, I.: Synergy of Interlaminar glass fiber hybridization on mechanical and dynamic characteristics of jute and flax fabric reinforced epoxy composites. J. Nat. Fibers 19, 4310–4325 (2022). https://doi.org/10.1080/15440478.2020.185628010.1080/15440478.2020.1856280 ↗
52. Bulut, M., Alsaadi, M., Erkliğ, A., Alrawi, H.: The effects of S-glass fiber hybridization on vibration-damping behavior of intraply woven carbon/aramid hybrid composites for different lay-up configurations. Proc. Inst. Mech. Eng. Part. C J. Mech. Eng. Sci. 233, 3220–3231 (2019). https://doi.org/10.1177/095440621881318810.1177/0954406218813188 ↗
53. Chen, C.S., Tsai, T.C., Chen, T.J., Chen, W.R.: Vibration and stability of initially stressed hybrid composite plates in hygrothermal environments. Mech. Compos. Mater. 53, 441–456 (2017). https://doi.org/10.1007/s11029-017-9674-810.1007/s11029-017-9674-8 ↗
54. Ebrahimi, F., Dabbagh, A.: On thermo-mechanical vibration analysis of multi-scale hybrid composite beams. J. Vib. Control. 25, 933–945 (2019). https://doi.org/10.1177/107754631880680010.1177/1077546318806800 ↗
55. Zhu, C., Prasad, M., Javeed Siddique, M., Ahsan, M., Margabandu, S., Al-Bahrani, M., Saeed, M., Selvaraj, A., Sivakumar, R., Yvaz, J.: Buckling and dynamic behavior of uniform and tapered woven carbon/jute fiber reinforced polyester hybrid composite beams. Mech. Adv. Mater. Struct. (2023). https://doi.org/10.1080/15376494.2023.221583710.1080/15376494.2023.2215837 ↗
56. Prasad, M., Maneengam, A., Siddique, M.J., Selvaraj, R.: Static and dynamic characteristics of jute/glass fiber reinforced hybrid composites. Structures 50, 954–962 (2023). https://doi.org/10.1016/j.istruc.2023.01.11610.1016/j.istruc.2023.01.116 ↗
57. Kallannavar, V., Kumaran, B., Kattimani, S.C.: Effect of temperature and moisture on free vibration characteristics of skew laminated hybrid composite and sandwich plates. Thin-Walled Struct. 157, 107113 (2020). https://doi.org/10.1016/j.tws.2020.10711310.1016/j.tws.2020.107113 ↗
58. Zhao, J., Xie, F., Wang, A., Shuai, C., Tang, J., Wang, Q.: Dynamics analysis of functionally graded porous (FGP) circular, annular and sector plates with general elastic restraints. Compos. Part B Eng. 159, 20–43 (2019). https://doi.org/10.1016/j.compositesb.2018.08.11410.1016/j.compositesb.2018.08.114 ↗
59. Wang, Q., Xie, F., Liu, T., Qin, B., Yu, H.: Free vibration analysis of moderately thick composite materials arbitrary triangular plates under multi-points support boundary conditions. Int. J. Mech. Sci. 184, 105789 (2020). https://doi.org/10.1016/j.ijmecsci.2020.10578910.1016/j.ijmecsci.2020.105789 ↗
60. Zuo, P., Shi, X., Ge, R., Luo, J.: Unified series solution for thermal vibration analysis of composite laminated joined conical-cylindrical shell with general boundary conditions. Thin-Walled Struct. 178, 109525 (2022). https://doi.org/10.1016/j.tws.2022.10952510.1016/j.tws.2022.109525 ↗
61. Li, H., Pang, F., Miao, X., Gao, S., Liu, F.: A semi analytical method for free vibration analysis of composite laminated cylindrical and spherical shells with complex boundary conditions. Thin-Walled Struct. 136, 200–220 (2019). https://doi.org/10.1016/j.tws.2018.12.00910.1016/j.tws.2018.12.009 ↗
62. Castro, S.G.P., Mittelstedt, C., Monteiro, F.A.C., Arbelo, M.A., Ziegmann, G., Degenhardt, R.: Linear buckling predictions of unstiffened laminated composite cylinders and cones under various loading and boundary conditions using semi-analytical models. Compos. Struct. 118, 303–315 (2014). https://doi.org/10.1016/j.compstruct.2014.07.03710.1016/j.compstruct.2014.07.037 ↗
63. Kouchakzadeh, M.A., Rahgozar, M., Bohlooly, M.: Buckling of laminated composite plates with elastically restrained boundary conditions. Struct. Eng. Mech. 74, 577–588 (2020)
64. Shao, D., Hu, S., Wang, Q., Pang, F.: Free vibration of refined higher-order shear deformation composite laminated beams with general boundary conditions. Compos. Part B Eng. 108, 75–90 (2017). https://doi.org/10.1016/j.compositesb.2016.09.09310.1016/j.compositesb.2016.09.093 ↗
65. Li, H., Hao, Y.X., Zhang, W., Liu, L.T., Yang, S.W., Wang, D.M.: Vibration analysis of porous metal foam truncated conical shells with general boundary conditions using GDQ. Compos. Struct. 269, 114036 (2021). https://doi.org/10.1016/j.compstruct.2021.11403610.1016/j.compstruct.2021.114036 ↗
66. Rostami, H., Jedari Salami, S.: Large amplitude free vibration of sandwich beams with flexible core and FG graphene platelet Reinforced Composite (FG-GPLRC) face sheets based on extended higher-order sandwich panel theory. Thin-Walled Struct. 180, 109999 (2022). https://doi.org/10.1016/j.tws.2022.10999910.1016/j.tws.2022.109999 ↗
67. Sun, S., Cao, D., Han, Q.: Vibration studies of rotating cylindrical shells with arbitrary edges using characteristic orthogonal polynomials in the Rayleigh–Ritz method. Int. J. Mech. Sci. 68, 180–189 (2013). https://doi.org/10.1016/j.ijmecsci.2013.01.01310.1016/j.ijmecsci.2013.01.013 ↗
68. Guo, C., Liu, T., Wang, Q., Qin, B., Shao, W., Wang, A.: Spectral-tchebychev technique for the free vibration analysis of composite laminated stepped and stiffened cylindrical shells with arbitrary boundary conditions. Compos. Struct. 272, 114193 (2021). https://doi.org/10.1016/j.compstruct.2021.11419310.1016/j.compstruct.2021.114193 ↗
69. Li, C., Li, P., Zhong, B., Miao, X.: Large-amplitude vibrations of thin-walled rotating laminated composite cylindrical shell with arbitrary boundary conditions. Thin-Walled Struct. 156, 106966 (2020). https://doi.org/10.1016/j.tws.2020.10696610.1016/j.tws.2020.106966 ↗
70. Li, H., Hao, Y.X., Zhang, W., Liu, L.T., Yang, S.W., Cao, Y.T.: Natural vibration of an elastically supported porous truncated joined conical-conical shells using artificial spring technology and generalized differential quadrature method. Aerosp. Sci. Technol. 121, 107385 (2022). https://doi.org/10.1016/j.ast.2022.10738510.1016/j.ast.2022.107385 ↗
71. Jin, G., Ye, T., Ma, X., Chen, Y., Su, Z., Xie, X.: A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 75, 357–376 (2013). https://doi.org/10.1016/j.ijmecsci.2013.08.00310.1016/j.ijmecsci.2013.08.003 ↗
72. Qin, B., Zhong, R., Wu, Q., Wang, T., Wang, Q.: A unified formulation for free vibration of laminated plate through Jacobi-Ritz method. Thin-Walled Struct. 144, 106354 (2019). https://doi.org/10.1016/j.tws.2019.10635410.1016/j.tws.2019.106354 ↗
73. Wang, Q., Cui, X., Qin, B., Liang, Q.: Vibration analysis of the functionally graded carbon nanotube reinforced composite shallow shells with arbitrary boundary conditions. Compos. Struct. 182, 364–379 (2017). https://doi.org/10.1016/j.compstruct.2017.09.04310.1016/j.compstruct.2017.09.043 ↗
74. Zhang, H., Shi, D., Zha, S., Wang, Q.: A simple first-order shear deformation theory for vibro-acoustic analysis of the laminated rectangular fluid-structure coupling system. Compos. Struct. 201, 647–663 (2018). https://doi.org/10.1016/j.compstruct.2018.06.09310.1016/j.compstruct.2018.06.093 ↗