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BY 4.0 license Open Access Published by De Gruyter Open Access May 29, 2019

Dynamic and Sound Radiation Characteristics of Rectangular Thin Plates with General Boundary Conditions

  • Yuan Du , Haichao Li , Qingtao Gong EMAIL logo , Fuzhen Pang and Liping Sun

Abstract

Based on the classical Kirchhoff hypothesis, the dynamic response and sound radiation of rectangular thin plates with general boundary conditions are studied. The transverse displacements of plate are represented by a double Fourier cosine series and three supplementary functions. The potential discontinuity associated with the original governing equation can be transferred to auxiliary series functions. All kinds of boundary conditions can be easily achieved by varying stiffness value of springs on each edge. The natural frequencies and vibration response of the plates are obtained by means of the Rayleigh–Ritz method. Sound radiation characteristics of the plate are derived using Rayleigh integral formula. Current method works well when handling dynamic response and sound radiation of plates with general boundary conditions. The accuracy and reliability of current method are confirmed by comparing with related literature and FEM. The non-dimensional frequency parameters of the rectangular plates with different boundary conditions and aspect ratios are presented in the paper, which may be useful for future researchers.Meanwhile, some interesting points are foundwhen analyzing acoustic radiation characteristics of plates.

References

[1] Liew, K.M., K.Y. Lam, and S.T. Chow, Free vibration analysis of rectangular plates using orthogonal plate function. Computers & Structures, 1990. 34(1): p. 79-85.10.1016/0045-7949(90)90302-ISearch in Google Scholar

[2] Wu, J.H., A.Q. Liu, and H.L. Chen, Exact Solutions for Free-Vibration Analysis of Rectangular Plates Using Bessel Functions. Journal of Applied Mechanics, 2007(6).10.1115/1.2744043Search in Google Scholar

[3] Takashi, M. and Y. Jin, Application of the collocation method to vibration analysis of rectangular mindlin plates. Computers & Structures, 1984. 18(3): p. 425-431.10.1016/0045-7949(84)90062-2Search in Google Scholar

[4] Liew, K.M. and T.M. Teo, THREE-DIMENSIONAL VIBRATION ANALYSIS OF RECTANGULAR PLATES BASED ON DIFFERENTIAL QUADRATURE METHOD. Journal of Sound and Vibration, 1999. 220(4): p. 577-599.10.1006/jsvi.1998.1927Search in Google Scholar

[5] Liu, G.R. and X.L. Chen, A MESH-FREE METHOD FOR STATIC AND FREE VIBRATION ANALYSES OF THIN PLATES OF COMPLICATED SHAPE. Journal of Sound and Vibration, 2001. 241(5): p. 839-855.10.1006/jsvi.2000.3330Search in Google Scholar

[6] Bert, C.W., S.K. Jang, and A.G. Striz, Two new approximate methods for analyzing free vibration of structural components. Aiaa Journal, 2015. 26(5): p. 612-618.10.2514/3.9941Search in Google Scholar

[7] Wanji, C. and Y.K. Cheung, Refined triangular discrete Kirchhoff plate element for thin plate bending, vibration and buckling analysis. International Journal for Numerical Methods in Engineering, 1998. 41(8): p. 1507-1525.10.1002/(SICI)1097-0207(19980430)41:8<1507::AID-NME351>3.0.CO;2-TSearch in Google Scholar

[8] Lim, C.W., et al., On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. International Journal of Engineering Science, 2009. 47(1): p. 131-140.10.1016/j.ijengsci.2008.08.003Search in Google Scholar

[9] Li, S. and H. Yuan, Green quasifunction method for free vibration of clamped thin plates. Acta Mechanica Solida Sinica, 2012. 25(1): p. 37-45.10.1016/S0894-9166(12)60004-4Search in Google Scholar

[10] Tornabene, F., 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution. Composite Structures, 2011. 93(7): p. 1854-1876.10.1016/j.compstruct.2011.02.006Search in Google Scholar

[11] Tornabene, F., N. Fantuzzi, and M. Bacciocchi, Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Composites Part B Engineering, 2014. 67(1): p. 490-509.10.1016/j.compositesb.2014.08.012Search in Google Scholar

[12] Tornabene, F., et al., Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation. Composites Part B Engineering, 2013. 55(1): p. 642-659.10.1016/j.compositesb.2013.07.026Search in Google Scholar

[13] Tornabene, F. and E. Viola, 2-D solution for free vibrations of parabolic shells using generalized differential quadrature method. European Journal of Mechanics - A/Solids, 2008. 27(6): p. 1001-1025.10.1016/j.euromechsol.2007.12.007Search in Google Scholar

[14] Tornabene, F. and E. Viola, Free vibration analysis of functionally graded panels and shells of revolution. Meccanica, 2009. 44(3): p. 255-281.10.1007/s11012-008-9167-xSearch in Google Scholar

[15] Kang, S.W. and S.N. Atluri, Improved non-dimensional dynamic influence function method for vibration analysis of arbitrarily shaped plates with clamped edges. Advances in Mechanical Engineering, 2016. 8(3).10.1177/1687814016638586Search in Google Scholar

[16] Li, X.K., J.F. Zhang, and Y. Zheng, Static and Free Vibration Analysis of Laminated Composite Plates Using Isogeometric Approach Based on the Third Order Shear Deformation Theory. Advances in Mechanical Engineering, 2014.10.1155/2014/232019Search in Google Scholar

[17] Thai, H.T. and D.H. Choi, A simple first-order shear deformation theory for the bending and free vibration analysis of functionally graded plates. Composite Structures, 2013. 101(15): p. 332-340.10.1016/j.compstruct.2013.02.019Search in Google Scholar

[18] Thai, H.T., T. Park, and D.H. Choi, An eflcient shear deformation theory for vibration of functionally graded plates. Archive of Applied Mechanics, 2013. 83(1): p. 137-149.10.1007/s00419-012-0642-4Search in Google Scholar

[19] Thai, H.T. and D.H. Choi, Analytical solutions of refined plate theory for bending, buckling and vibration analyses of thick plates. Applied Mathematical Modelling, 2013. 37(18-19): p. 8310-8323.10.1016/j.apm.2013.03.038Search in Google Scholar

[20] Thai, H.T., M. Park, and D.H. Choi, A simple refined theory for bending, buckling, and vibration of thick plates resting on elastic foundation. International Journal of Mechanical Sciences, 2013. 73(73): p. 40-52.10.1016/j.ijmecsci.2013.03.017Search in Google Scholar

[21] Thai, H.T. and S.E. Kim, Free vibration of laminated composite plates using two variable refined plate theory. International Journal of Mechanical Sciences, 2010. 52(4): p. 626-633.10.1016/j.ijmecsci.2010.01.002Search in Google Scholar

[22] Dozio, L. and M. Ricciardi, Free vibration analysis of ribbed plates by a combined analytical–numerical method. Journal of Sound & Vibration, 2009. 319(1): p. 681-697.10.1016/j.jsv.2008.06.024Search in Google Scholar

[23] Dozio, L., On the use of the Trigonometric Ritz method for general vibration analysis of rectangular Kirchhoff plates. Thin-Walled Structures, 2011. 49(1): p. 129-144.10.1016/j.tws.2010.08.014Search in Google Scholar

[24] Dozio, L., Natural frequencies of sandwich plates with FGM core via variable-kinematic 2-D Ritz models. Composite Structures, 2013. 96: p. 561-568.10.1016/j.compstruct.2012.08.016Search in Google Scholar

[25] Jayasinghe, S. and S.M. Hashemi, A Dynamic Coeflcient Matrix Method for the Free Vibration of Thin Rectangular Isotropic Plates. Shock and Vibration, 2018.10.1155/2018/1071830Search in Google Scholar

[26] Li, H.C., et al., An Accurate Solution Method for the Static and Vibration Analysis of Functionally Graded Reissner-Mindlin Rectangular Plate with General Boundary Conditions. Shock and Vibration, 2018: p. 21.10.1155/2018/4535871Search in Google Scholar

[27] Shi, S.X., et al., Modeling and Simulation of Transverse Free Vibration Analysis of a Rectangular Plate with Cutouts Using Energy Principles. Shock and Vibration, 2018: p. 16.10.1155/2018/9609745Search in Google Scholar

[28] Allahverdizadeh, A., M.H. Naei, and M.N. Bahrami, Nonlinear free and forced vibration analysis of thin circular functionally graded plates. Journal of Sound & Vibration, 2008. 310(4): p. 966-984.10.1016/j.jsv.2007.08.011Search in Google Scholar

[29] Han, W. and M. Petyt, Linear vibration analysis of laminated rectangular plates using the hierarchical finite element method— II. Forced vibration analysis. Computers&Structures, 1996. 61(4): p. 713-724.10.1016/0045-7949(96)00213-1Search in Google Scholar

[30] Akay, A., M. Tokunaga, and M. Latcha, A theoretical analysis of transient sound radiation from a clamped circular plate. American Society of Mechanical Engineers, 1984. 51(1): p. 41-47.10.1115/1.3167595Search in Google Scholar

[31] Srinivas, S. and A.K. Rao, Bending, vibration and buckling of simply supported thick orthotropic rectangular plates and laminates. International Journal of Solids & Structures, 1970. 6(11): p. 1463-1481.10.1016/0020-7683(70)90076-4Search in Google Scholar

[32] Shi, D.Y., et al., Free and Forced Vibration of the Moderately Thick Laminated Composite Rectangular Plate on Various Elastic Winkler and Pasternak Foundations. Shock and Vibration, 2017: p. 23.10.1155/2017/7820130Search in Google Scholar

[33] Inalpolat, M., M. Caliskan, and R. Singh, Analysis of near field sound radiation from a resonant unbaffled plate using simplified analytical models. Noise Control Engineering Journal, 2010. 58(2): p. 145-156.10.3397/1.3322184Search in Google Scholar

[34] Mace, B.R., Sound radiation from a plate reinforced by two sets of parallel stiffeners. Journal of Sound and Vibration, 1980. 71(3): p. 435-441.10.1016/0022-460X(80)90425-3Search in Google Scholar

[35] Laulagnet, B., Sound radiation by a simply supported unbaffled plate. Journal of the Acoustical Society of America, 1998. 103(5): p. 2451-2462.10.1121/1.422765Search in Google Scholar

[36] Sorokin, S.V., Vibrations of and sound radiation from sandwich plates in heavy fluid loading conditions. Composite Structures, 2000. 48(4): p. 219-230.10.1016/S0263-8223(99)00103-8Search in Google Scholar

[37] Li, H., et al., Vibration analysis of functionally graded porous cylindrical shell with arbitrary boundary restraints by using a semi analytical method. Composites Part B: Engineering, 2019. 164: p. 249-264.10.1016/j.compositesb.2018.11.046Search in Google Scholar

[38] Pang, F., et al., A modified Fourier solution for vibration analysis of moderately thick laminated annular sector plates with general boundary conditions, internal radial line and circumferential arc supports. Curved and Layered Structures, 2017. 4(1): p. 189-220.10.1515/cls-2017-0014Search in Google Scholar

[39] Li, H., F. Pang, and H. Chen, A semi-analytical approach to analyze vibration characteristics of uniform and stepped annularspherical shells with general boundary conditions. European Journal of Mechanics - A/Solids, 2019. 74: p. 48-65.10.1016/j.euromechsol.2018.10.017Search in Google Scholar

[40] Pang, F., et al., Application of flügge thin shell theory to the solution of free vibration behaviors for spherical-cylindrical-spherical shell: A unified formulation. European Journal of Mechanics - A/Solids, 2019. 74: p. 381-393.10.1016/j.euromechsol.2018.12.003Search in Google Scholar

[41] Li, H., et al., Free vibration analysis of uniform and stepped combined paraboloidal, cylindrical and spherical shells with arbitrary boundary conditions. International Journal of Mechanical Sciences, 2018. 145: p. 64-82.10.1016/j.ijmecsci.2018.06.021Search in Google Scholar

[42] Li, H., et al., A semi analytical method for free vibration analysis of composite laminated cylindrical and spherical shells with complex boundary conditions. Thin-Walled Structures, 2019. 136: p. 200-220.10.1016/j.tws.2018.12.009Search in Google Scholar

[43] Li, H., et al., Application of first-order shear deformation theory for the vibration analysis of functionally graded doubly-curved shells of revolution. Composite Structures, 2019. 212: p. 22-42.10.1016/j.compstruct.2019.01.012Search in Google Scholar

[44] Li, H., et al., Jacobi–Ritz method for free vibration analysis of uniform and stepped circular cylindrical shells with arbitrary boundary conditions: A unified formulation. Computers & Mathematics with Applications, 2018.10.1016/j.camwa.2018.09.046Search in Google Scholar

[45] Pang, F., et al., Free vibration of functionally graded carbon nanotube reinforced composite annular sector plate with general boundary supports. Curved and Layered Structures, 2018. 5(1): p. 49-67.10.1515/cls-2018-0005Search in Google Scholar

[46] Pang, F., et al., Application of First-Order Shear Deformation Theory on Vibration Analysis of Stepped Functionally Graded Paraboloidal Shell with General Edge Constraints. Materials, 2018. 12(1): p. 69.10.3390/ma12010069Search in Google Scholar PubMed PubMed Central

[47] Pang, F., et al., Free vibration analysis of combined composite laminated cylindrical and spherical shells with arbitrary boundary conditions. Mechanics of Advanced Materials and Structures, 2019: p. 1-18.10.1080/15376494.2018.1553258Search in Google Scholar

[48] Li, W.L., et al., An exact series solution for the transverse vibration of rectangular plateswith general elastic boundary supports. Journal of Sound & Vibration, 2009. 321(1): p. 254-269.Search in Google Scholar

[49] Li, H., et al., Free vibration analysis for composite laminated doubly-curved shells of revolution by a semi analytical method. Composite Structures, 2018. 201: p. 86-111.10.1016/j.compstruct.2018.05.143Search in Google Scholar

[50] Pang, F., et al., A semi analytical method for the free vibration of doubly-curved shells of revolution. Computers & Mathematics with Applications, 2018. 75(9): p. 3249-3268.10.1016/j.camwa.2018.01.045Search in Google Scholar

[51] Pang, F., et al., Free and Forced Vibration Analysis of Airtight Cylindrical Vessels with Doubly Curved Shells of Revolution by Using Jacobi-Ritz Method. Shock and Vibration, 2017. 2017.10.1155/2017/4538540Search in Google Scholar

[52] Li, H., et al., A semi-analytical method for vibration analysis of stepped doubly-curved shells of revolution with arbitrary boundary conditions. Thin-Walled Structures, 2018. 129: p. 125-144.10.1016/j.tws.2018.03.026Search in Google Scholar

Received: 2018-12-11
Accepted: 2019-01-24
Published Online: 2019-05-29
Published in Print: 2019-01-01

© 2019 Yuan Du et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 Public License.

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