DOI QR코드

DOI QR Code

On mixing the Rayleigh-Ritz formulation with Hankel's function for vibration of fluid-filled functionally graded cylindrical shell

  • Hussain, Muzamal (Department of Mathematics, Govt. College University Faisalabad) ;
  • Naeem, Muhammad Nawaz (Department of Mathematics, Govt. College University Faisalabad) ;
  • Shahzad, Aamir (Department of Physics, Govt. College University Faisalabad) ;
  • Taj, Muhammad (Department of Mathematics, University of Azad Jammu and Kashmir) ;
  • Asghar, Sehar (Department of Mathematics, Govt. College University Faisalabad) ;
  • Fatahi-Vajari, Alireza (Department of Mechanical Engineering, Shahryar Branch, Islamic Azad University) ;
  • Singh, Rahul (University Department of Mechanical Engineering, Rajasthan Technical University) ;
  • Tounsi, Abdelouahed (Materials and Hydrology Laboratory University of Sidi Bel Abbes, Algeria Faculty of Technology Civil Engineering Department)
  • 투고 : 2019.12.31
  • 심사 : 2020.02.13
  • 발행 : 2020.10.25

초록

In this paper, a cylindrical shell is immersed in a non-viscous fluid using first order shell theory of Sander. These equations are partial differential equations which are solved by approximate technique. Robust and efficient techniques are favored to get precise results. Employment of the Rayleigh-Ritz procedure gives birth to the shell frequency equation. Use of acoustic wave equation is done to incorporate the sound pressure produced in a fluid. Hankel's functions of second kind designate the fluid influence. Mathematically the integral form of the Lagrange energy functional is converted into a set of three partial differential equations. Throughout the computation, simply supported edge condition is used. Expressions for modal displacement functions, the three unknown functions are supposed in such way that the axial, circumferential and time variables are separated by the product method. Comparison is made for empty and fluid-filled cylindrical shell with circumferential wave number, length- and height-radius ratios, it is found that the fluid-filled frequencies are lower than that of without fluid. To generate the fundamental natural frequencies and for better accuracy and effectiveness, the computer software MATLAB is used.

키워드

참고문헌

  1. 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.
  2. Amabili, M., Pellicano, F. and Paidoussis, M.P. (1998), "Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid", J. Fluids Struct., 12(7), 883-918. https://doi.org/10.1006/jfls.1999.0225.
  3. 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.
  4. Arshad, S.H., Naeem, M.N. and Sultana, N. (2007), "Frequency analysis of functionally graded cylindrical shells with various volume fraction laws", J. Mech. Eng. Sci., 221, 1483-1495. https://doi.org/10.1243/09544062JMES738.
  5. Asghar, S., Naeem, M.N. and Hussain, 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.
  6. Avcar, M. (2019), "Free vibration of imperfect sigmoid and power law functionally graded beams", Steel Composite Struct., 30(6), 603-606. https://doi.org/10.12989/scs.2019.30.6.603.
  7. Chen, Y., Zhao, H.B. and Shea, Z.P. (1993), "Vibrations of high-speed rotating shells with calculations for cylindrical shells", J. Sound Vib., 160, 137-160. https://doi.org/10.1006/jsvi.1993.1010.
  8. Chi, S.H. and Chung, Y.L. (2006), "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.
  9. Chung, H., Turula, P., Mulcahy, T.M. and Jendrzejczyk, J.A. (1981), "Analysis of cylindrical shell vibrating in a cylindrical fluid region", Nucl. Eng. Des., 63(1), 109-120. https://doi.org/10.1016/0029-5493(81)90020-0.
  10. Dong, S.B. (1977), "A block-stodola eigen solution technique for large algebraic systems with nonsymmetrical matrices", Int. J. Number Methods Eng., 11, 247-267. https://doi.org/10.1002/nme.1620110204.
  11. 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.
  12. Ersoy, H., Mercan, K. and Civalek, O. (2018), "Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods", Composite Struct., 183, 7-20. https://doi.org/10.1016/j.compstruct.2016.11.051.
  13. Farahani, H. and Barati, F. (2015), "Vibration of submerged functionally graded cylindrical shell based on first order shear deformation theory using wave propagation method", Struct. Eng. Mech., 53(3), 575-587. https://doi.org/10.12989/sem.2015.53.3.575.
  14. Fatahi-Vajari, A., Azimzadeh, Z. and Hussain, M., (2019), "Nonlinear coupled axial-torsional vibration of single-walled carbon nanotubes using Galerkin and homotopy perturbation method", Micro Nano Lett., 14(14), 1366-1371. https://doi.org/10.1049/mnl.2019.0203.
  15. Flügge, W. (1962), Stresses in shells, Springer-Verlag, Berlin.
  16. Gasser, L.F.F. (1987), "Free vibrations on thin cylindrical shells containing liquid", M.Sc. Dissertation, Federal University of Rio de Janerio, Rio de Janerio, Portugal.
  17. 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/10.1016/0022-460X(88)90354-9.
  18. Goncalves, P.B., Da Silva, F.M.A. and Prado, Z.J.G.N. (2006), "Transient stability of empty and fluid-filled cylindrical shells", J. Braz. Soc. Mech. Sci. Eng., 28(3), 331-333. http://dx.doi.org/10.1590/S1678-58782006000300011.
  19. Gonçalves, P.B., Silva, F. and del Prado, Z.J. (2006), "Transient stability of empty and fluid-filled cylindrical shells international symposium on dynamic problems of mechanics, J. Brazilian Society Mech. Sci. Eng., 28(3), 331-338. https://doi.org/10.1590/S1678-58782006000300011.
  20. Hussain, M, Naeem, M.N. and Taj. M. (2019b), "Vibration characteristics of zigzag and chiral FGM rotating carbon nanotubes sandwich with ring supports", Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 233(16), 5763-5780. https://doi.org/10.1177/0954406219855095.
  21. Hussain, M. and Naeem, M.N. (2017), "Vibration analysis of single-walled carbon nanotubes using wave propagation approach", Mech. Sci., 8(1),155-164. https://doi.org/10.5194/ms-8-155-2017.
  22. Hussain, M. and Naeem, M.N. (2018a), Advance Testing and Engineering, Intechopen, London, U.K.
  23. Hussain, M. and Naeem, M.N. (2018b), Novel Nanomaterials: Synthesis and Applications, Intechopen, London, U.K. https://doi.org/10.5772/intechopen.73503.
  24. Hussain, M. and Naeem, M.N. (2019a), "Effects of ring supports on vibration of armchair and zigzag FGM rotating carbon nanotubes using Galerkin's method", Compos. Part B Eng., 163, 548-561. https://doi.org/10.1016/j.compositesb.2018.12.144.
  25. Hussain, M. and Naeem, M.N. (2019b), "Rotating response on the vibrations of functionally graded zigzag and chiral single walled carbon nanotubes", Appl. Math. Model., 75, 506-520. https://doi.org/10.1016/j.apm.2019.05.039.
  26. Hussain, M. and Naeem, M.N. (2020), "Mass density effect on vibration of zigzag and chiral SWCNTs", J. Sandw. Struct. Mater., 1099636220906257. https://doi.org/10.1177/1099636220906257.
  27. Hussain, M., Naeem, M., Shahzad, A. and He, M. (2017), "Vibrational behavior of single-walled carbon nanotubes based on cylindrical shell model using wave propagation approach", AIP Adv., 7(4), 045114. https://doi.org/10.1063/1.4979112.
  28. Hussain, M., Naeem, M.N and Tounsi, A. (2020), "Simulating vibration of single-walled carbon nanotube based on Relagh-Ritz Method", Adv. Nano Res., 8(3), 215-228. https://doi.org/10.12989/anr.2020.8.3.215.
  29. Hussain, M., Naeem, M.N. and Isvandzibaei, M. (2018a), "Effect of Winkler and Pasternak elastic foundation on the vibration of rotating functionally graded material cylindrical shell", Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 232(24), 4564-4577. https://doi.org/10.1177/0954406217753459.
  30. Hussain, M., Naeem, M.N. and Taj, M. (2019c), "Effect of length and thickness variations on the vibration of SWCNTs based on Flügge's shell model", Micro Nano Lett., 15(1), 1-6. https://doi.org/10.1049/mnl.2019.0309.
  31. Hussain, M., Naeem, M.N., Shahzad, A., He, M. and Habib, S. (2018b), "Vibrations of rotating cylindrical shells with functionally graded material using wave propagation approach", Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 232(23), 4342-4356. https://doi.org/10.1177/0954406218802320.
  32. Hussain, M., Naeem, M.N., Tounsi, A. and Taj, M. (2019a), "Nonlocal effect on the vibration of armchair and zigzag SWCNTs with bending rigidity", Adv. Nano Res., 7(6), 431-442. https://doi.org/10.12989/anr.2019.7.6.431.
  33. 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.
  34. Khayat, M., Dehghan, S.M., Najafgholipour, M.A. and Baghlani, A. (2018), "Free vibration analysis of functionally graded cylindrical shells with different shell theories using semi-analytical method", Steel Compos. Struct., 28(6), 735-748. https://doi.org/10.12989/scs.2018.28.6.735.
  35. Koizumi, M. (1997), "FGM activities in Japan", Compos. Part B, 28(1-2), 1-4. https://doi.org/10.1016/S1359-8368(96)00016-9.
  36. Lam, K.Y. and Loy, C.T. (1998), "Influence of boundary conditions for a thin laminated rotating cylindrical shell", Compos. Struct., 41, 215-228. https://doi.org/10.1016/S0263-8223(98)00012-9.
  37. Li, H., Pang, F., Du, Y. and Gao, C. (2019), "Free vibration analysis of uniform and stepped functionally graded circular cylindrical shells", Steel Compos. Struct., 33(2), 163-180. https://doi.org/10.12989/scs.2019.33.2.163.
  38. Love, A.E.H. (1888), "The small free vibrations and deformation of a thin elastic shell", Philos. Trans. R. Soc. Lond. B Biol. Sci., 179, 491-546. 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.L. and Shu, C. (1997), "Analysis of cylindrical shells using generalized differential quadrature", Shock Vib., 4(3), 193-198. https://doi.org/10.3233/SAV-1997-4305.
  41. Loy, C.T., Lam, K.Y. and Reddy, J.N. (1999), "Vibration of functionally graded cylindrical shells", Int. J. Mech. Sci., 41, 309-324. https://doi.org/10.1016/S0020-7403(98)00054-X.
  42. Mercan, K., Demir, C. and Civalek, O. (2016), "Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique", Curved Layer. Struct., 3(1), 0007. https://doi.org/10.1515/cls-2016-0007.
  43. 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.
  44. 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 Mech., 191, 75-91. https://doi.org/10.1007/s00707-006-0438-0.
  45. Orsberg, K. (1964), "Influence of boundary conditions on modal characteristics of cylindrical shells", J. Amer. Institute Aeronautics Astronautics, 2, 182-189.
  46. Rahimi, G.H., Ansari, R. and Hemmatnezhad, M. (2011), "Vibration of functionally graded cylindrical shells with ring support", Sci. Iran., 18(6), 1313-1320. https://doi.org/10.1016/j.scient.2011.11.026.
  47. Sehar, A., Hussain, M., Naeem M.N. and Tounsi, A. (2020), "Prediction and assessment of nonlocal natural frequencies DWCNTs: Vibration analysis", Comput. Concrete, 25(2), 133-144. https://doi.org/10.12989/cac.2020.25.2.133.
  48. Sewall, J.L. and Naumann, E.C. (1968), An Experimental and Analytical Vibration Study of Thin Cylindrical Shells with and without Longitudinal Stiffeners, NASA, Washington D.C., U.S.A.
  49. Shah, A.G., Mahmood, T. and Naeem, M.N. (2009), "Vibrations of FGM thin cylindrical shells with exponential volume fraction law", Appl. Math. Mech., 30(5), 607-615. https://doi.org/10.1007/s10483-009-0507-x.
  50. Sharma, C.B. and Johns, D.J. (1971), "Vibration characteristics of a clamped-free and clamped-ringstiffened circular cylindrical shell", J. Sound Vib., 14(4), 459-474. https://doi.org/10.1016/0022-460X(71)90575-X.
  51. 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.
  52. Sharma, P., Singh, R. and Hussain, M. (2019), "On modal analysis of axially functionally graded material beam under hygrothermal effect", Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 234(5), 1085-1101. https://doi.org/10.1177/0954406219888234.
  53. Sodel, W. (1981), Vibration of Shell and Plates, Mechanical Engineering Series, New York, U.S.A.
  54. 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.2010.2403.
  55. Sofiyev, A.H., Alizada, A.N., Akin, O., Valiyev, A., Avcar, M. and Adiguzel, S. (2012), "On the stability of FGM shells subjected to combined loads with different edge conditions and resting on elastic foundations", Acta Mech., 223(1), 189-204. https://doi.org/10.1007/s00707-011-0548-1.
  56. Suresh, S. and Mortensen, A. (1997), "Functionally gradient metals and metal ceramic composites part 2: Thermo mechanical behavior", Int. Mater. Rev., 42(3), 85-116. https://doi.org/10.1179/imr.1997.42.3.85.
  57. Toulokian, Y.S. (1967), Thermophysical Properties of High Temperature Solid Materials, Macmillan, New York, U.S.A.
  58. 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.
  59. 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).
  60. 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.
  61. Wuite, J. and Adali, S. (2005), "Deflection and stress behavior of nanocomposite reinforced beams using a multiscale analysis", Compos. Struct., 71(3-4), 388-96. https://doi.org/10.1016/j.compstruct.2005.09.011.
  62. 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.
  63. 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.
  64. Zhang, X.M. (2002), "Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation approach", Comput. Methods Appl. Mech. Eng., 191, 2057-2071. https://doi.org/10.1016/S0045-7825(01)00368-1.
  65. 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.