Browse > Article
http://dx.doi.org/10.12989/was.2019.28.6.347

Conformable solution of fractional vibration problem of plate subjected to in-plane loads  

Fadodun, Odunayo O. (Department of Mathematics, Obafemi Awolowo University)
Malomo, Babafemi O. (Department of Mechanical Engineering, Obafemi Awolowo University)
Layeni, Olawanle P. (Department of Mathematics, Obafemi Awolowo University)
Akinola, Adegbola P. (Department of Mathematics, Obafemi Awolowo University)
Publication Information
Wind and Structures / v.28, no.6, 2019 , pp. 347-354 More about this Journal
Abstract
This study provides an approximate analytical solution to the fractional vibration problem of thin plate governing anomalous motion of plate subjected to in-plane loads. The method of variable separable is employed to transform the fractional partial differential equations under consideration into a fractional ordinary differential equation in temporal variable and a bi-harmonic plate equation in spatial variable. The technique of conformable fractional derivative is utilized to solve the resulting fractional differential equation and the approach of finite sine integral transform method is used to solve the accompanying bi-harmonic plate equation. The deflection field which measures the transverse displacement of the plate is expressed in terms of product of Bessel and trigonometric functions via the temporal and spatial variables respectively. The obtained solution reduces to the solution of the free vibration problem of thin plate in literature. This work shows that conformable fractional derivative is an efficient mathematical tool for tracking analytical solution of fractional partial differential equation governing anomalous vibration of thin plates.
Keywords
conformable solution; fractional vibration; in-plane loads; thin plate;
Citations & Related Records
Times Cited By KSCI : 15  (Citation Analysis)
연도 인용수 순위
1 Fadodun, O.O. and Akinola, A.P. (2017), "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.   DOI
2 Fadodun, O.O., Borokinni, A.S., Layeni, O.P. and Akinola, A.P. (2017a), "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.   DOI
3 Fadodun, O.O., Layeni, O.P. and Akinola, A.P. (2017b), "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.   DOI
4 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.   DOI
5 Hadji, L., Zouatnia, N. and Kassoul, A. (2017), "Wave propagation in functionally graded beams using various higher-order shear deformation beams theories", Struct. Eng. Mech., 62(2), 143-149. https://doi.org/10.12989/sem.2017.62.2.143.   DOI
6 Hadji, L., Khelifa, Z. and Adda Bedia, E.A. (2016), "A new higher-ordershear deformation model for functionally graded beams", J. Civil Eng.- KSCE, 20(5), 1835-1841.   DOI
7 Hebali, H., Tounsi, A., Houari, M.S.A. and Bessaim, A. (2014), "A new quasi-3D hyperbolic shear deformation theory for the static and free vibration analysis of functionally graded plates", J. Eng. Mech.- ASCE, 140(2), 374-383.   DOI
8 Jaroszewick J. (2017), "Natural frequencies of axisymmetric vibration of thin hyperbolic circular plates with clamped conditions", Int. J. Appl. Mech. Eng., 22(2), 451-457. DOI: 10.1515/ijame-2017-0028.   DOI
9 Khalil, R., Al Horani, M., Yousef, A. and Sababheh, M. (2014), "A new definition of fractional derivative", J. Comput. Appl. Mech., 264(1), 65-70. https://doi.org/10.1016/j.cam.2014.01.002   DOI
10 Abdelaziz, H.H., Meziane, M.A.A., Bousahla, A.A., Tounsi, A., Mahmoud, S.R. and Alwabli, A.S. (2017), "An efficient hyperbolic shear deformation theory for bending, buckling and free vibration of FGM sandwich plates with various boundary conditions", Steel Compos. Struct., 25(6), 693-704. https://doi.org/10.12989/scs.2017.25.6.693.   DOI
11 Abualnour, M., Houari, M.S.A., Tounsi, A., Bedia, E.A.A. and Mahmoud, S.R. (2018), "A novel quasi-3D trigonometric plate theory for free vibration analysis of advanced composite plates", Compos. Struct., 184, 688-697. https://doi.org/10.1016/j.compstruct.2017.10.047.   DOI
12 Altekin, M. (2017), "Free transverse vibration of shear deformable super-elliptical plates", Wind Struct., 24(4), 307-331. https://doi.org/10.12989/was.2017.24.4.307.   DOI
13 An, C., Gu, J. and Su, J. (2015), "Exact solution of bending of clamped orthotropic rectangular thin plates", J. Braz. Soc. Mech. Sci. Eng., DOI 10.1007/s40430-0329-1.
14 Bakhadda, B., Bouiadjra, M.B., Bourada, F., Bousahla, A.A., Tounsi, A. and Mahmoud, S.R. (2018), "Dynamic and bending analysis of carbon nanotube-reinforced composite plates with elastic foundation", Wind Struct., 27(5), 311-324. https://doi.org/10.12989/was.2018.27.5.311.   DOI
15 Bao, S. and Wang, S. (2017), "A generalized solution procedure for in-plane free vibration of rectangular plates and annular sectorial plates", Roy. Soc. Open Science, 4(8), doi: 10.1098/rsos.170484.
16 Lal, R. and Saini, R. (2016), "Mode shapes and frequencies of thin rectangular plates with arbitrarily varying non-homogeneity along two concurrent edge", J. Vib. Control, 23(17), 2841-2865. https://doi.org/10.1177/1077546315623710.   DOI
17 Li, X. (2014), "Analytical solution to a fractional generalized two phase Lame-Clapeyron Stefan problem", Int. J. Numer. Method. Heat. Fluid Fl., 24(6), 1251-1259.   DOI
18 Lindsay, A.E., Hao, W. and Sommese, A.J. (2015), "Vibrations of thin plates with small clamped patches", Proceedings of the Royal Society A, doi: 10.1098/rspa.2015.0474.
19 Liu, M.F. and Chang, T.P. (2010), "Closed form expression for the vibration of a transversely isotropic magneto-electro-elastic plate", J. Appl. Mech. T. - ASME, 77, doi: 10.1115/1.3176996.
20 Lychev S.A., Lycheva, T.N. and Manzhirov, A.V. (2011), "Unsteady vibration of a growing circular plate", Mech. Solids, 46(2), 325-333.   DOI
21 Park, I., Lee, U. and Park, D. (2015), "Transverse vibration of the thin plates: Frequency-domain spectral element modeling and analysis", Math. Problem Eng., doi: 10.1155/2015/541276.
22 Rao, S.S. (2007), "Vibration of continuous system", John Willey and Sons, Inc, Hobokean, New Jersey, U.S.A.
23 Senjanovic, I., Hadzic, N. and Vladimir, N. (2015), "Vibration analysis of thin circular plates with multiple openings by the assumed mode method", Proceeding of the Institution of Mechanical Engineers, Part M: Journal of Engineering for the Maritime Environment, 231(1), 70-85.   DOI
24 Shooshtari, A. and Razavi, S. (2015), "Nonlinear vibration analysis of rectangular magneto-electro-elastic thin plates", IJE T. A, 28 (1), 136-144. doi: 10.5829/idosi.ije.2015.28.01a.18.
25 Bouafia, K.A., Houari, M.S.A., Benzair, A. and Tounsi, A. (2017), "A nonlocal quasi-3D theory for bending and free flexural vibration behaviors of functionally graded nanobeams", Smart Struct. Syst., 19(2), 115-126. https://doi.org/10.12989/sss.2017.19.2.115.   DOI
26 Belabed, Z., Bousahla, A.A., Houari, M.S.A., Tounsi, A. and Mahmoud, S.R. (2018), "A new 3-unknown hyperbolic shear deformation theory for vibration of functionally graded sandwich plate", Earthq. Struct., 14(2), 103-115. https://doi.org/10.12989/eas.2018.14.2.103.   DOI
27 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.   DOI
28 Berferhat, R., Daouadji, T.H., Mansour, M.S. 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.   DOI
29 Bouhadra, A., Tounsi, A., Bousahla, A.A., Benyoucef, S., Mahmoud, S.R. (2018), "Improved HSDT accounting for effect of thickness stretching in advanced composite plates", Struct. Eng. Mech., 66(1), 61-73. https://doi.org/10.12989/sem.2018.66.1.061.   DOI
30 Bounouara, F., Benrahou, K.H., Belkorissat, I. and Tounsi, A., (2016), "A nonlocal zeroth-order shear deformation theory for free vibration of functionally graded nanoscale plates resting on elastic foundation", Steel Compos. Struct., 20(2), 227-249. https://doi.org/10.12989/scs.2016.20.2.227.   DOI
31 Bourada, F., Bousahla, A.A., Bourada, M., Azzaz, A., Zinata, A. and Tounsi, A. (2019), "Dynamic investigation of porous functionally graded beam using a sinusoidal shear deformation theory", Wind Struct., 28(1), 19-30. https://doi.org/10.12989/was.2019.28.1.019.   DOI
32 Younsi, A., Tounsi, A. and Zhora, Z.F. (2018), "Novel quasi-3D and 2D shear deformation theories for bending and free vibration analysis of FGM plates", Geomech. Eng., 14(6), 519-532. https://doi.org/10.12989/gae.2018.14.6.519.   DOI
33 Tahouneh V. (2018), "3-D Vibration analysis of FGMWCNTs/Phenolic sandwich sectorial plates", Steel Compos. Struct., 26 (5), 649-662. https://doi.org/10.12989/scs.2018.26.5.649.   DOI
34 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.   DOI
35 Ventsel, E. and Krauthammer, T. (2001), "Thin plate and shell theory, analysis and application", Marce Dekker, Inc., New York and Basel NY, USA.
36 Zemri, A., Houari, M.S.A., Bousahla, A.A. and Tounsi, A. (2015), "A mechanical response of functionally graded nanoscale beam: an assessment of a refined nonlocal shear deformation theory beam theory", Struct. Eng. Mech., 54(4), 693-710. https://doi.org/10.12989/sem.2015.54.4.693.   DOI
37 Zhong, Y., Zhao, X. and Liu, H. (2014), "Vibration of plate on foundation with four edges free by finite cosine integral transform method", Latin Am. J. Solids Struct., 11(5), 854-863. http://dx.doi.org/10.1590/S1679-78252014000500008.   DOI
38 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.   DOI
39 Bourada, F., Amara, K. and Tounsi, A. (2016), "Buckling analysis of isotropic and orthotropic plates using a novel four variable refined plate theory", Steel Compos. Struct., 21(6), 1287-1306. http://dx.doi.org/10.12989/scs.2016.21.6.1287.   DOI
40 Chen, W., Ye, L.J. and Sun, H.G. (2010), "Fractional diffusion equation by the Kansa method", Comput. Mathod. Appl. M., 59, 1614-1620. https://doi.org/10.1016/j.camwa.2009.08.004.