DOI QR코드

DOI QR Code

Analysis of axisymmetric fractional vibration of an isotropic thin disc in finite deformation

  • Fadodun, Odunayo O. (Department of Mathematics, Obafemi Awolowo University)
  • 투고 : 2018.06.14
  • 심사 : 2019.04.08
  • 발행 : 2019.05.25

초록

This study investigates axisymmetric fractional vibration of an isotropic hyperelastic semi-linear thin disc with a view to examine effects of finite deformation associated with the material of the disc and effects of fractional vibration associated with the motion of the disc. The generalized three-dimensional equation of motion is reduced to an equivalent time fraction one-dimensional vibration equation. Using the method of variable separable, the resulting equation is further decomposed into second-order ordinary differential equation in spatial variable and fractional differential equation in temporal variable. The obtained solution of the fractional vibration problem under consideration is described by product of one-parameter Mittag-Leffler and Bessel functions in temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem in literature. Finally, and amongst other things, the Cauchy's stress distribution in thin disc under finite deformation exhibits nonlinearity with respect to the displacement fields whereas in infinitesimal deformation hypothesis, these stresses exhibit linear relation with the displacement field.

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참고문헌

  1. Akinola, A.P. (1999), "An energy function for transversely isotropic elastic material and Ponyting effect", J. Appl. Math. Comput., 6(3), 639-649. https://doi.org/10.1007/BF03009956.
  2. Akinola, A.P. (2001), "An application of nonlinear fundamental problems of a transversely isotropic layer in finite elastic deformation", Int. J. Nonlin. Mech., 36(2), 307-321. https://doi.org/10.1016/S0020-7462(00)00016-0.
  3. Bashmal, S., Bhat, R. and Rakheja, S., (2010), "Frequency equations for the in-plane vibration of circular annular disks", Adv. Acoust. Vib., 2010, Article ID 501902, 8. http://dx.doi.org/10.1155/2010/501902.
  4. Batra, R.C. and Iaccarino, G.L. (2008), "Exact solutions for radial deformations of a functionally-graded isotropic and incompressible second order elastic cylinder", Int. J. Nonlin. Mech., 43(5), 383-398. https://doi.org/10.1016/j.ijnonlinmec.2008.01.006.
  5. Benferhat, R., Tahar, H.D., Said-Mansour, M. and Hadji, L. (2016), "Effect of porosity on the bending and free vibration response of functionally graded plates resting on Winkler-Pasternak foundations", Earthq. Struct., 10(6), 1429-1449. https://doi.org/10.12989/eas.2016.10.6.1429.
  6. Bennoun, M., Houari, M.S.A. and Tounsi, A. (2016), "A novel five variable refined plate theory for vibration analysis of functionally graded sandwich plates", Mech. Adv. Mater. Struct., 23(4), 423-431. https://doi.org/10.1080/15376494.2014.984088.
  7. Bouboulas, A.S. and Anifantis, N.K. (2011), "Vibration analysis of a rotating disk with crack", Mech. Eng., 2011, Article ID 727120, 13. doi:10.5402/2011/727120.
  8. Burago, N.G., Nikitin, A.D., Nikitin, I.S. and Yushkovsky, P.A. (2016), "Stationary vibrations and fatigue failure of compressor disks of variable thickness", Procedia Struct. Integrity: 21st Eur. Conf. Fract., ECF 21, 2, 1109-1116. https://doi.org/10.1016/j.prostr.2016.06.142.
  9. Chen, W., Ye, L. and Sun, H. (2010), "Fractional diffusion equations by the Kansa method", Comput. Math. Appl., 59(5), 1614-1620. https://doi.org/10.1016/j.camwa.2009.08.004.
  10. Ciarlet, P.G. (1998), Mathematical Elasticity Volume I: Three-Dimensional Elasticity, Elsevier Science Publisher, Amsterdam.
  11. Das, D., Sahoo, P. and Saha, K. (2010), "Free vibration analysis of a rotating annular disc under uniform pressure loading", Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 224(3), 615-634. https://doi.org/10.1243/09544062JMES1662.
  12. Deng, H. and Ouyang, H., (2010), "Vibration of spinning discs and powder formation in centrifugal atomization", Proc. R. Soc. A: Math. Phys. Eng. Sci., 467(2126), 361-380. https://doi.org/10.1098/rspa.2010.0099.
  13. Du, R., Cao, W.R. and Sun, Z.Z. (2010), "A compact difference scheme for the fractional diffusion-wave equation", Appl. Math. Model., 34(10), 2998-3007. https://doi.org/10.1016/j.apm.2010.01.008.
  14. Fadodun, O.O. and Akinola, A.P. (2017a), "Bending of an isotropic non-classical thin rectangular plate", Struct. Eng. Mech., 61(4), 437-440. https://doi.org/10.12989/sem.2017.61.4.437.
  15. Fadodun, O.O., Borokinni, A.S., Layeni, O.P. and Akinola, A.P. (2017b), "Dynamics analysis of a transversely isotropic nonclassical thin plate", Wind Struct., 25(1), 25-38. https://doi.org/10.12989/was.2017.25.1.025.
  16. Fadodun, O.O., Layeni, O.P. and Akinola, A.P. (2017c), "Fractional wave propagation in radially vibrating non-classical cylinder", Earthq. Struct., 13(5), 465-471. https://doi.org/10.12989/eas.2017.13.5.465.
  17. Fu, Z.J., Chen, W. and Yang, H.T. (2013), "Boundary particle method for Laplace transformed time fractional diffusion equations", J. Comput. Phys., 235, 52-66. https://doi.org/10.1016/j.jcp.2012.10.018.
  18. Gorman, D.G., Reese, J.M., Horacek, J. and Dedouch, K. (2001), "Vibration analysis of a circular disc backed by a cylindrical cavity", Proc. Inst. Mech. Eng., Part C: J. Mech. Eng. Sci., 215(11), 1303-1311. https://doi.org/10.1243/0954406011524685.
  19. Hasheminejad, S.M., Ghaheri, A. and Vaezian, S. (2013), "Exact solution for free in-plane vibration analysis of an eccentric elliptical plate", Acta Mechanica, 224(8), 1609-1624. https://doi.org/10.1007/s00707-013-0829-y.
  20. Hutton, D.V. (2004), Fundamentals of Finite Element Analysis, Mc Graw Hill.
  21. Jaroszewick J. (2017), "Natural frequencies of axisymmetric vibration of thin hyperbolic circular plates with clamped edges", Int. J. Appl. Mech. Eng., 22(2), 451-457. DOI: 10.1515/ijame- 2017-0028.
  22. Kumar, R., Reen, L.S. and Garg, S.K. (2017), "Effects of time and diffusion phase-lags in a thin circular disc with axisymmetric heat supply", Cogent Math., 4(1), 1369848. doi.org/10.1080/23311835.2017.1369848.
  23. Li, X. (2014), "Analytical solutions to a fractional generalized two phase Lame-Clapeyron Stefan problem", Int. J. Numer. Meth. Heat. Fluid Flow, 24(6), 1251-1259. https://doi.org/10.1108/HFF-03-2013-0102.
  24. Lychev, S.A, Lycheva, T.N. and Manzhirov, A.V. (2011), "Unsteady vibration of a growing circular plate", Mech. Solid., 46(2), 325-333. https://doi.org/10.3103/S002565441102021X.
  25. Lyu, P., Du, J., Liu, Z. and Zhang, P. (2017), "Free in-plane vibration analysis of elastically restrained annular panels made of functionally graded material", Compos. Struct., 178(15), 246-259. https://doi.org/10.1016/j.compstruct.2017.06.065.
  26. Senjanovic, I., Hadzic, N. and Vladimir, N. (2015), "Vibration analysis of thin circular plates with multiple openings by the assumed mode method", Proc. Inst. Mech. Eng., Part M: J. Eng. Maritime Environ., 231(1), 70-85. https://doi.org/10.1177/1475090215621578.
  27. Sharma, J.N., Sharma, D. and Kumar, S. (2012), "Stress and strain analysis of rotating FGM thermoelastic circular disk by using FEM", Int. J. Pure Appl. Math., 73(3), 339-352.
  28. Treeby, B.E. and Cox, B.T. (2010), "Modeling power law absorption and dispersion for acoustic propagation using the fractional Laplacian", J. Acoust. Soc. Am., 195(5), 2741-2748. https://doi.org/10.1121/1.3377056.
  29. Ursoniu, C., Pepa, D., Tufoi, M. and Gillich G.R. (2017), "The influence of stiffening ribs on the natural frequencies of butterfly valve disks", Int. Conf. Appl. Sci.: Mater. Sci. Eng., 163(1), 012041. doi:10.1088/1757-899X/163/1/012041.
  30. Zhang, H., Yuan, H., Yang, W. and Zhao, T. (2017), "Research on vibration localization of mistuned bladed disk system", J. Vibroeng., 19(5), 3296-3312. https://doi.org/10.21595/jve.2017.17822.
  31. Zhong, R., Wang, Q., Tang, J., Shuai, C. and Qin, B. (2018), "Vibration analysis of functionally graded carbon nanotube reinforced composites (FG-CNTRC) circular, annular and sector plates", Compos. Struct., 194(15), 49-67. https://doi.org/10.1016/j.compstruct.2018.03.104.
  32. Zur, K.K. (2015), "Green's function in frequency analysis of circular thin plates of variable thickness", J. Theor. Appl. Mech., 53(4), 873-884. doi: 10.15632/jtam-pl.53.4.873.
  33. Zur, K.K. (2016a), "Green's function approach to frequency analysis of thin circular plates", Bull. Polish Acad. Sci. Tech. Sci., 64(1), 181-188. DOI: 10.1515/bpasts-2016-0020.
  34. Zur, K.K. (2016b), "Green's function for frequency analysis of thin annular plates with nonlinear variable thickness", Appl. Math. Model., 40(5-6), 3601-3619. https://doi.org/10.1016/j.apm.2015.10.014.
  35. Zur, K.K. (2018), "Quasi-Green's function approach to free vibration analysis of elastically supported functionally graded circular plates", Compos. Struct., 183(1), 600-610. https://doi.org/10.1016/j.compstruct.2017.07.012.

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