Smart Structures and Systems

  • 제27권5호
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  • Pages.729-743
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  • 2021
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  • 1738-1584(pISSN)
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  • 1738-1991(eISSN)
테크노프레스 (Techno-Press)
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Vibration analysis of FG cylindrical shell: Evaluation of Ritz-polynomial mixed with ring terms

  • Khadimallah, Mohamed A. (Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department) ;
  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad) ;
  • Naeem, Muhammad Nawaz (Department of Mathematics, Govt. College University Faisalabad) ;
  • Qazaq, Amjad (Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department) ;
  • Alqahtani, Abdulaziz (Prince Sattam Bin Abdulaziz University, College of Engineering, Civil Engineering Department) ;
  • Tounsi, Abdelouahed (YFL (Yonsei Frontier Lab), Yonsei University)
  • 투고 : 2020.06.11
  • 심사 : 2021.01.02
  • 발행 : 2021.05.25
⟨ 이전 논문 다음 논문 ⟩

초록

Here the Rayleigh - Ritz method has been applied to derive the shell vibration frequency equation. This equation has been formed as an eigenvalue problem form. MATLAB software package has been utilized for extracting shell frequency spectra. Nature of materials used for construction of cylindrical shells also has visible impact on shell vibration characteristics. For isotropic materials, the physical properties are same everywhere, the laminated and functionally graded materials vary from point to point. Here the shell material has been taken as functionally graded material. Moreover, the impact of ring supports around the shell circumferential has been examined for the various positions along the shell axial length. These shells are stiffened by rings in the tangential direction. These ring supports are located at various positions along the axial direction round the shell circumferential direction. These variations have been plotted against the locations of ring supports for three values of exponents of volume fraction law. For three conditions, frequency variations show different behavior with these values of exponent law. The influence of the positions of ring supports for simply supported end conditions is very visible. The frequency first increases and gain maximum value in the midway of the shell length and then lowers down. The comparisons of frequencies have been made for efficiency and robustness for the present numerical procedure.

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참고문헌

  1. Alazzawy, W.I. (2008), "Static and Dynamic Analysis of Stiffened Plate Used in Machine Tool Column", J. Eng., 14(4), 3099-3111.
  2. Alazzawy, W.I. and Jweeg, M.J. (2010), "A study of free vibration and fatigue for cross-ply closed cylindrical shells using General Third shell Theory (GTT)", J. Eng., 16(2), 5170-5184.
  3. AlSaleh, R.J. and Fuggini, C. (2020), "Combining GPS and accelerometers' records to capture torsional response of cylindrical tower", Smart Struct. Syst., Int. J., 25(1), 111-122. https://doi.org/10.12989/sss.2020.25.1.111
  4. Amabili, M. (1999), "Vibration of circular tubes and shells filled and partially immersed in dense fluids", J. Sound Vib., 221(4), 567-585. https://doi.org/10.1006/jsvi.1998.2050
  5. Amabili, M., Pellicano, F. and Paidoussis, M.P. (1998), "Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid", 12(7), 883-918. https://doi.org/10.1006/jfls.1998.0173
  6. Ansari, R. and Rouhi, H. (2015), "Nonlocal Flugge shell model for the axial buckling of single-walled Carbon nanotubes: An analytical approach", Int. J. Nano Dimens., 6(5), 453-462. https://doi.org/10.7508/IJND.2015.05.002
  7. Arani, A.G., Kolahchi, R. and Esmailpour, M. (2016), "Nonlinear vibration analysis of piezoelectric plates reinforced with carbon nanotubes using DQM", Smart Struct. Syst., Int. J., 18(4), 787-800. https://doi.org/10.12989/sss.2016.18.4.787
  8. Arefi, M. and Zenkour, A.M. (2017), "Nonlinear and linear thermo-elastic analyses of a functionally graded spherical shell using the Lagrange strain tensor", Smart Struct. Syst., Int. J., 19(1), 33-38. https://doi.org/10.12989/sss.2017.19.1.033
  9. Arnold, R.N. and Warburton, G.B. (1949), "Flexural vibrations of the walls of thin cylindrical shells having freely supported ends", Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 197(1049), 238-256. http://dx.doi.org/10.1098/rspa.1949.0061
  10. Asghar, S., Hussain, M. and Naeem, M. (2019), "Non-local effect on the vibration analysis of double walled carbon nanotubes based on Donnell shell theory", Physica E: Low-dimens. Syst. Nanostruct., 116, 113726. https://doi.org/10.1016/j.physe.2019.113726
  11. Bisen, H.B., Hirwani, C.K., Satankar, R.K., Panda, S.K., Mehar, K. and Patel, B. (2018), "Numerical study of frequency and deflection responses of natural fiber (Luffa) reinforced polymer composite and experimental validation", J. Natural Fibers, 1-15. https://doi.org/10.1080/15440478.2018.1503129
  12. Boussoula, A., Boucham, B., Bourada, M., Bourada, F., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2019), "A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates", Smart Struct. Syst., Int. J., 25(2), 197-218. https://doi.org/10.12989/sss.2020.25.2.197
  13. Boussoula, A., Boucham, B., Bourada, M., Bourada, F., Tounsi, A., Bousahla, A.A. and Tounsi, A. (2020), "A simple nth-order shear deformation theory for thermomechanical bending analysis of different configurations of FG sandwich plates", Smart Struct. Syst., Int. J., 25(2), 197-218. https://doi.org/10.12989/sss.2020.25.2.197
  14. Chi, S.H. and Chung, Y.L. (2006a), "Mechanical behavior of functionally graded material plates under transverse load-Part I: Analysis", Int. J. Solids Struct., 43(13), 3657-3674. https://doi.org/10.1016/j.ijsolstr.2005.04.010
  15. Chi, S.H. and Chung, Y.L. (2006b), "Mechanical behavior of functionally graded material plates under transverse load-part II: numerical results", Int. J. Solids Struct., 43, 3657-3691. https://doi.org/10.1016/j.ijsolstr.2005.04.010
  16. Chung, H., Turula, P., Mulcahy, T.M. and Jendrzejczyk, J.A. (1981), "Analysis of cylindrical shell vibrating in a cylindrical fluid region", Nuclear Eng. Des., 63(1), 109-1012. https://doi.org/10.1016/0029-5493(81)90020-0
  17. Dewangan, H.C., Panda, S.K. and Sharma, N. (2020a), "Experimental Validation of Role of Cut-Out Parameters on Modal Responses of Laminated Composite-A Coupled Fe Approach", Int. J. Appl. Mech., 12(6), 2050068. https://doi.org/10.1142/S1758825120500684
  18. Dewangan, H.C., Sharma, N., Hirwani, C.K. and Panda, S.K. (2020b), "Numerical eigenfrequency and experimental verification of variable cutout (square/rectangular) borne layered glass/epoxy flat/curved panel structure", Mech. Based Des. Struct. Mach., 1-18. https://doi.org/10.1080/15397734.2020.1759432
  19. Dong, S.B. (1977), "A block-stodola eigen solution technique for large algebraic systems with non-symmetrical matrices", Int. J. Numer. Methods Eng., 11, 247. https://doi.org/10.1002/nme.1620110204
  20. Ergin, A. and Temarel, P. (2002), "Free vibration of a partially liquid-filled and submerged, horizontal cylindrical shell", J. Sound Vib., 254(5), 951-965. https://doi.org/10.1006/jsvi.2001.4139
  21. Flugge, W. (1962), Stresses in Shells, (2nd Edition), Springer-Verlag, Berlin, Germany.
  22. Flugge, W. (1967), Stresses in Shells, (2nd Edition), Springer-Verlag, Berlin, Germany. https://www.springer.com/gp/book/9783662010280
  23. Forsberg, K. (1964), "Influence of boundary conditions on modal characteristics of cylindrical shells", J. Am. Inst. Aeronaut. Astronaut., 2, 182-189. https://arc.aiaa.org/doi/abs/10.2514/6.1964-77
  24. Galletly, G.D. (1955), "On the in-vacuo vibrations of simply supported, ring-stiffened cylindrical shells", US National Congress of Applied Mechanics. http//www.vacuo-vibrations-supported-ring-stiffened-cylindrical/dp/B0007FTWBQ
  25. Gasser, L.F.F. (1987), "Free vibrations on thin cylindrical shells containing liquid", M.S. Thesis, Federal University of Rio de Janerio, peccoppe-ufrj, Rio de Janerio, Portugal. [In Portuguese]
  26. Ghobaei-Arani, M., Jabbehdari, S. and Pourmina, M.A. (2018), "An autonomic resource provisioning approach for service-based cloud applications: A hybrid approach", Future Gener. Comput. Syst., 78, 191-210. https://doi.org/10.1016/j.future.2017.02.022
  27. Goncalves, P.B. and Batista, R.C. (1987), "Frequency response of cylindrical shells partially submerged or filled with liquid", J. Sound Vib., 113(1), 59-70. https://doi.org/10.1016/S0022-460X(87)81340-8
  28. Goncalves, P.B. and Batista, R.C. (1988), "Non-linear vibration analysis of fluid-filled cylindrical shells", J. Sound Vib., 127(1), 133-143. https://doi.org/10.1006/jsvi.2001.4139
  29. Jiang, J. and Olson, M.D. (1994), "Vibrational analysis of orthogonally stiffened cylindrical shells using super elements", J. Sound Vib., 173, 73-83. https://doi.org/10.1006/jsvi.1994.1218
  30. Jweeg, M.J. and Alazzawy, W.I. (2007), "A suggested analytical solution for laminated closed cylindrical shells using General Third Shell Theory (GTT)", Al-Nahrain J. Eng. Sci., 10(1), 11-26.
  31. Jweeg, M.J. and Majeed, W.I. (2020), "Free vibration Analysis solution for laminated truncated conical shells using high orde theory", Proceedings of the 6th Sc Conference of the College of Engineering, University of Baghdad, Volume 3, pp. 208-225.
  32. Kareem, M.G. and Majeed, W.I. (2019), "Transient dynamic analysis of laminated shallow spherical shell under low-velocity impact", J. Mater. Res. Technol., 8(6), 5283-5300. https://doi.org/10.1016/j.jmrt.2019.08.050
  33. Koizumi, M. (1997), "FGM Activities in Japan", Composites. https://doi.org/10.1016/S1359-8368(96)00016-9
  34. Krommer, M., Vetyukova, Y. and Staudigl, E. (2016), "Nonlinear modelling and analysis of thin piezoelectric plates: buckling and post-buckling behavior", Smart Struct. Syst., Int. J., 18(1), 155-181. https://doi.org/10.12989/sss.2016.18.1.155
  35. Lam, K.Y. and Loy, C.T. (1998), "Influence of boundary conditions for a thin laminated rotating cylindrical shell", Compos. Struct., 41(3-4), 215-228. https://doi.org/10.1016/S0263-8223(98)00012-9
  36. Lee, S.Y., Huynh, T.C., Dang, N.L. and Kim, J.T. (2019), "Vibration characteristics of caisson breakwater for various waves, sea levels, and foundations", Smart Struct. Syst., Int. J., 24(4), 525-539. https://doi.org/10.12989/sss.2019.24.4.525
  37. Leissa, A.W. (1973), "Vibration of shells". https://ntrs.nasa.gov/search.jsp?R=19730018197
  38. Love, A.E.H. (1888), "XVI. The small small free vibrations and deformation of thin elastic shell", Phil. Trans. R. Soc. London, A179, 491-549. https://doi.org/10.1098/rsta.1888.0016
  39. Loy, C.T. and Lam, K.Y. (1997), "Vibration of cylindrical shells with ring supports", J. Mech. Eng., 39, 455-471. https://doi.org/10.1016/S0020-7403(96)00035-5
  40. Loy, C.T., Lam, K.Y. and Reddy, J.N. (1999), "Vibration of functionally graded cylindrical shells", Int. J. Mech. Sci., 41(3), 309-324. https://doi.org/10.1016/S0020-7403(98)00054-X
  41. Marcel Dekker, Books: https://www.abebooks.com/book-search/title/vibrations-shells-plates/author/soedel-werner/
  42. Mehar, K. and Panda, S.K. (2016a), "Geometrical nonlinear free vibration analysis of FG-CNT reinforced composite flat panel under uniform thermal field", Compos. Struct., 143, 336-346. https://doi.org/10.1016/j.compstruct.2016.02.038
  43. Mehar, K. and Panda, S.K. (2016b), "Free vibration and bending behaviour of CNT reinforced composite plate using different shear deformation theory", Proceedings of IOP Conference Series: Materials Science and Engineering, 115(1), 012014.
  44. Mehar, K. and Panda, S.K. (2018a), "Dynamic response of functionally graded carbon nanotube reinforced sandwich plate", Proceedings of IOP Conference Series: Materials Science and Engineering, Vol. 338, No. 1, p. 012017.
  45. Mehar, K. and Panda, S.K. (2018b), "Thermal free vibration behavior of FG-CNT reinforced sandwich curved panel using finite element method", Polym. Compos., 39(8), 2751-2764. https://doi.org/10.1002/pc.24266
  46. Mehar, K. and Panda, S.K. (2018c), "Elastic bending and stress analysis of carbon nanotube-reinforced composite plate: Experimental, numerical, and simulation", Adv. Polym. Technol., 37(6), 1643-1657. https://doi.org/10.1002/adv.21821
  47. Mehar, K. and Panda, S.K. (2018d), "Thermoelastic flexural analysis of FG-CNT doubly curved shell panel", Aircr. Eng. Aerosp. Technol., 90(1), 11-23. https://doi.org/10.1108/AEAT-11-2015-0237
  48. Mehar, K. and Panda, S.K. (2018e), "Nonlinear finite element solutions of thermoelastic flexural strength and stress values of temperature dependent graded CNT-reinforced sandwich shallow shell structure", Struct. Eng. Mech., Int. J., 67(6), 565-578. https://doi.org/10.12989/sem.2018.67.6.565
  49. Mehar, K. and Panda, S.K. (2019), "Multiscale modeling approach for thermal buckling analysis of nanocomposite curved structure", Adv. Nano Res., Int. J., 7(3), 181-190. https://doi.org/10.12989/anr.2019.7.3.181
  50. Mehar, K., Panda, S.K., Dehengia, A. and Kar, V.R. (2016), "Vibration analysis of functionally graded carbon nanotube reinforced composite plate in thermal environment", J. Sandw. Struct. Mater., 18(2), 151-173. https://doi.org/10.1177/1099636215613324
  51. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017a), "Thermoelastic nonlinear frequency analysis of CNT reinforced functionally graded sandwich structure", Eur. J. Mech.- A/Solids, 65, 384-396. https://doi.org/10.1016/j.euromechsol.2017.05.005
  52. Mehar, K., Panda, S.K., Bui, T.Q. and Mahapatra, T.R. (2017b), "Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM", J. Thermal Stress., 40(7), 899-916. https://doi.org/10.1080/01495739.2017.1318689
  53. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2017c), "Theoretical and experimental investigation of vibration characteristic of carbon nanotube reinforced polymer composite structure", Int. J. Mech. Sci., 133, 319-329. https://doi.org/10.1016/j.ijmecsci.2017.08.057
  54. Mehar, K., Panda, S.K. and Patle, B.K. (2017d), "Thermoelastic vibration and flexural behavior of FG-CNT reinforced composite curved panel", Int. J. Appl. Mech., 9(4), 1750046. https://doi.org/10.1142/S1758825117500466
  55. Mehar, K., Mahapatra, T.R., Panda, S.K., Katariya, P.V. and Tompe, U.K. (2018a), "Finite-element solution to nonlocal elasticity and scale effect on frequency behavior of shear deformable nanoplate structure", J. Eng. Mech., 144(9), 04018094. https://doi.org/10.1061/(ASCE)EM.1943-7889.0001519
  56. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018b), "Thermoelastic deflection responses of CNT reinforced sandwich shell structure using finite element method", Scientia Iranica, 25(5), 2722-2737. https://doi.org/10.24200/SCI.2017.4525
  57. Mehar, K., Panda, S.K. and Patle, B.K. (2018c), "Stress, deflection, and frequency analysis of CNT reinforced graded sandwich plate under uniform and linear thermal environment: A finite element approach", Polym. Compos., 39(10), 3792- 3809. https://doi.org/10.1002/pc.24409
  58. Mehar, K., Panda, S.K. and Mahapatra, T.R. (2018d), "Nonlinear frequency responses of functionally graded carbon nanotube-reinforced sandwich curved panel under uniform temperature field", Int. J. Appl. Mech., 10(3), 1850028. https://doi.org/10.1142/S175882511850028X
  59. Mehar, K., Panda, S.K., Devarajan, Y. and Choubey, G. (2019), "Numerical buckling analysis of graded CNT-reinforced composite sandwich shell structure under thermal loading", Compos. Struct., 216, 406-414. https://doi.org/10.1016/j.compstruct.2019.03.002
  60. Moazzez, K., Googarchin, H.S. and Sharifi, S.M.H. (2018), "Natural frequency analysis of a cylindrical shell containing a variably oriented surface crack utilizing line-spring model", Thin-Wall. Struct., 125, 63-75. https://doi.org/10.1016/j.tws.2018.01.009
  61. Naeem, M.N., Ghamkhar, M., Arshad, S.H. and Shah, A.G. (2013), "Vibration analysis of submerged thin FGM cylindrical shells", J. Mech. Sci. Technol., 27(3), 649-656. https://doi.org/10.1007/s12206-013-0119-6
  62. Najafizadeh, M.M. and Isvandzibaei, M.R. (2007), "Vibration of (FGM) cylindrical shells based on higher order shear deformation plate theory with ring support", Acta Mechanica, 191, 75-91. https://doi.org/10.1007/s00707-006-0438-0
  63. Poplawski, B., Mikulowski, G., Pisarski, D., Wiszowaty, R. and Jankowski, L. (2019), "Optimum actuator placement for damping of vibrations using the Prestress-Accumulation Release control approach", Smart Struct. Syst., Int. J., 24(1), 27-35. https://doi.org/10.12989/sss.2019.24.1.027
  64. Ramteke, P., Mehar, K., Sharma, N. and Panda, S. (2020a), "Numerical Prediction of Deflection and Stress Responses of Functionally Graded Structure for Grading Patterns (Power-Law, Sigmoid and Exponential) and Variable Porosity (Even/Uneven)", Scientia Iranica.
  65. Ramteke, P.M., Mahapatra, B.P., Panda, S.K. and Sharma, N. (2020b), "Static deflection simulation study of 2D Functionally graded porous structure", Materials Today: Proceedings, 33, 5544-5547. https://doi.org/10.1016/j.matpr.2020.03.537
  66. Sadoughifar, A., Farhatnia, F., Izadinia, M. and Talaeetaba, S.B. (2020), "Size-dependent buckling behaviour of FG annular/circular thick nanoplates with porosities resting on Kerr foundation based on new hyperbolic shear deformation theory", Struct. Eng. Mech., Int. J., 73(3), 225-238. https://doi.org/10.12989/sem.2020.73.3.225
  67. Sewall, J.L. and Naumann, E.C. (1968), "An experimental and analytical vibration study of thin cylindrical shells with and without longitudinal stiffeners", National Aeronautic and Space Administration; for sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. https://ntrs.nasa.gov/search.jsp?R=19680024266%202020-06-07T18:48:40+00:00Z
  68. Shah, A.G., Mahmood, T. and Naeem, M.N. (2009), "Vibrations of FGM thin cylindrical shells with exponential volume fraction law", Appl. Mathe. Mech., 30(5), 607-615. https://doi.org/10.1007/s10483-009-0507-x
  69. Sharma, C.B. (1974), "Calculation of natural frequencies of fixed-free circular cylindrical shells", J. Sound Vib., 35(1), 55-76. https://doi.org/10.1016/0022-460X(74)90038-8
  70. Sharma, C.B. and Johns, D.J. (1971), "Vibration characteristics of a clamped-free and clamped-ring-stiffened circular cylindrical shell", J. Sound Vib., 14(4), 459-474. https://doi.org/10.1016/0022-460X(71)90575-X
  71. Sharma, C.B., Darvizeh, M. and Darvizeh, A. (1998), "Natural frequency response of vertical cantilever composite shells containing fluid", Eng. Struct., 20(8), 732-737. https://doi.org/10.1016/S0141-0296(97)00102-8
  72. Sharma, P., Singh, R. and Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proceedings of the Institution of Mechanical Engineers, Part C: J. Mech. Eng. Sci., 234(5), 1085-1101. https://doi.org/10.1177/0954406219888234.
  73. Sodel, W. (1981), "Vibration of shell and plates", In: Mechanical Engineering Series, New York, USA.
  74. Sofiyev, A.H. and Avcar, M. (2010), "The stability of cylindrical shells containing an FGM layer subjected to axial load on the Pasternak foundation", Eng., 2, 228-236. https://doi.org/10.4236/eng.2010.24033
  75. Suresh, S. and Mortensen, A. (1997), "Functionally graded metals and metal-ceramic composites: Part 2 Thermomechanical behaviour", Int. Mater. Rev., 42, 85-116. https://doi.org/10.1179/imr.1995.40.6.239
  76. Swaddiwudhipong, S., Tian, J. and Wang, C.M. (1995), "Vibration of cylindrical shells with ring supports", J. Sound Vib., 187(1), 69-93. https://doi.org/10.1006/jsvi.1995.0503
  77. Tohidi, H., Hosseini-Hashemi, S.H. and Maghsoudpour, A. (2018), "Size-dependent forced vibration response of embedded micro cylindrical shells reinforced with agglomerated CNTs using strain gradient theory", Smart Struct. Syst., Int. J., 22(5), 527-546. https://doi.org/10.12989/sss.2018.22.5.527
  78. Toulokian, Y.S. (1967), Thermo Physical Properties of High Temperature Solid Materials, New York: Macmillan. https://apps.dtic.mil/dtic/tr/fulltext/u2/649947.pdf
  79. Wang, C. and Lai, J.C.S. (2000), "Prediction of natural frequencies of finite length circular cylindrical shells", Appl. Acoust., 59(4), 385-400. https://doi.org/10.1016/S0003-682X(99)00039-0
  80. Wang, C.M., Swaddiwudhipong, S. and Tian, J. (1997), "Ritz method for vibration analysis of cylindrical shells with ring-stiffeners", J. Eng. Mech., 123, 134-143. https://doi.org/10.1061/(ASCE)0733-9399(1997)123:2(134)
  81. Warburton, G.B. (1965), "Vibration of thin cylindrical shells", J. Mech. Eng. Sci., 7, 399-407. https://doi.org/10.1243/JMES_JOUR_1965_007_062_02
  82. Wuite, J. and Adali, S. (2005), "Deflection and stress behavior of nanocomposite reinforced beams using a multiscale analysis", Compos. Struct., 71(3-4), 388-396. https://doi.org/10.1016/j.compstruct.2005.09.011
  83. Xiang, Y., Ma, Y.F., Kitipornchai, S. and Lau, C.W.H. (2002), "Exact solutions for vibration of cylindrical shells with intermediate ring supports", Int. J. Mech. Sci., 44(9), 1907-1924. https://doi.org/10.1016/S0020-7403(02)00071-1
  84. Xuebin, L. (2008), "Study on free vibration analysis of circular cylindrical shells using wave propagation", J. Sound Vib., 311, 667-682. https://doi.org/10.1016/j.jsv.2007.09.023
  85. Yeh, J.Y. (2016), "Vibration characteristic analysis of sandwich cylindrical shells with MR elastomer", Smart Struct. Syst., Int. J., 18(2), 233-247. https://doi.org/10.12989/sss.2016.18.2.233
  86. Zahrai, S.M. and Kakouei, S. (2019), "Shaking table tests on a SDOF structure with cylindrical and rectangular TLDs having rotatable baffles", Smart Struct. Syst., Int. J., 24(3), 391-401. https://doi.org/10.12989/sss.2019.24.3.391
  87. Zhang, X.M. (2002), "Frequency analysis of submerged cylindrical shells with the wave propagation approach", J. Mech. Sci., 44, 1259-1273. https://doi.org/10.1016/S0020-7403(02)00059-0
  88. Zhang, X.M., Liu, G.R. and Lam, K.Y. (2001), "Coupled vibration of fluid-filled cylindrical shells using the wave propagation approach", Appl. Acoust., 62, 229-243. https://doi.org/10.1016/S0003-682X(00)00045-1