Nonlinear vibrations of a composite circular plate with a rigid body

doi:10.1007/s10483-023-3005-8

Abstract

Abstract: The influence of weights is usually ignored in the study of nonlinear vibrations of plates. In this paper, the effect of structure weights on the nonlinear vibration of a composite circular plate with a rigid body is presented. The nonlinear governing equations are derived from the generalized Hamilton's principle and the von Kármán plate theory. The equilibrium configurations due to weights are determined and validated by the finite element method (FEM). A nonlinear model for the vibration around the equilibrium configuration is established. Moreover, the natural frequencies and amplitude-frequency responses of harmonically forced vibrations are calculated. The study shows that the structure weights introduce additional linear and quadratic nonlinear terms into the dynamical model. This leads to interesting phenomena. For example, considering weights increases the natural frequency. Furthermore, when the influence of weights is considered, the vibration response of the plate becomes asymmetrical.

Key words: composite circular plate, weight, nonlinear vibration, equilibrium configuration, natural frequency

2010 MSC Number:

74H45
74K20

Ying MENG, Xiaoye MAO, Hu DING, Liqun CHEN. Nonlinear vibrations of a composite circular plate with a rigid body. Applied Mathematics and Mechanics (English Edition), 2023, 44(6): 857-876.

TrendMD

References

[1] ZHOU, S. M., SHENG, L. P., and SHEN, Z. B. Transverse vibration of circular graphene sheetbased mass sensor via nonlocal Kirchhoff plate theory. Computational Materials Science, 86, 73–78(2014)
[2] MARTIN, P. A. and HULL, A. J. Dynamic response of an infinite thin plate loaded with concentrated masses. Wave Motion, 98, 102643(2020)
[3] HU, Y. L., LIANG, X., and WANG, W. A theoretical solution of resonant circular diaphragm-type piezoactuators with added mass loads. Sensors and Actuators A: Physical, 258, 74–87(2017)
[4] YANG, Y. Y. W., WANG, S., STEIN, P., XU, B. X., and YANG, T. Q. Vibration-based energy harvesting with a clamped piezoelectric circular diaphragm: analysis and identification of optimal structural parameters. Smart Materials and Structures, 26(4), 045011(2017)
[5] NAYFEH, A. H. and MOOK, D. T. Nonlinear Oscillations, John Wiley and Sons, New York (1979)
[6] SATHYAMOORTHY, M. Nonlinear vibration analysis of plates: review and survey of current developments. Applied Mechanics Reviews, 40(11), 1553–1561(1987)
[7] AMABILI, M. Nonlinear Vibrations and Stability of Shells and Plates, Cambridge University Press, Cambridge (2008)
[8] CIVALEK, Ö. Geometrically nonlinear dynamic and static analysis of shallow spherical shell resting on two-parameters elastic foundations. International Journal of Pressure Vessels and Piping, 113, 1–9(2014)
[9] THOMSON, W. and DAHLEH, M. Theory of Vibration with Applications, 5th ed., Prentice Hall, Upper Saddle River (1998)
[10] MEIROVITCH, L. Fundamentals of Vibrations, McGraw-Hill, Boston (2001)
[11] ZHANG, Y. X. and YANG, C. H. Recent developments in finite element analysis for laminated composite plates. Composite Structures, 88(1), 147–157(2009)
[12] KAZANCI, Z. A review on the response of blast loaded laminated composite plates. Progress in Aerospace Sciences, 81, 49–59(2016)
[13] MAJI, A. and MAHATO, P. K. Development and applications of shear deformation theories for laminated composite plates: an overview. Journal of Thermoplastic Composite Materials, 35(12), 2576–2619(2020)
[14] HAMMAMI, M., El MAHI, A., KARRA, C., and HADDAR, M. Experimental analysis of the linear and nonlinear behaviour of composites with delaminations. Applied Acoustics, 108, 31–39(2016)
[15] LU, T. Y., CHEN, X. H., WANG, H., ZHANG, L., and ZHOU, Y. H. A novel experiment-based approach on nonlinear bending analysis of thermoplastic composite laminates. Composites Part B: Engineering, 209, 108611(2021)
[16] ARANI, A. G. and JAFARI, G. S. Nonlinear vibration analysis of laminated composite Mindlin micro/nano-plates resting on orthotropic Pasternak medium using DQM. Applied Mathematics and Mechanics (English Edition), 36(8), 1033–1044(2015) https://doi.org/10.1007/s10483-015-1969-7
[17] SOLANKI, M. K., KUMAR, R., and SINGH, J. Flexure analysis of laminated plates using multiquadratic RBF based meshfree method. International Journal of Computational Methods, 15(6), 1850049(2018)
[18] ESMAEILZADEH, M., KADKHODAYAN, M., MOHAMMADI, S., and TURVEY, G. J. Nonlinear dynamic analysis of moving bilayer plates resting on elastic foundations. Applied Mathematics and Mechanics (English Edition), 41(3), 439–458(2020) https://doi.org/10.1007/s10483-020-2587-8
[19] ZHOU, Y. and ZHANG, W. Double Hopf bifurcation of composite laminated piezoelectric plate subjected to external and internal excitations. Applied Mathematics and Mechanics (English Edition), 38(5), 689–706(2017) https://doi.org/10.1007/s10483-017-2196-9
[20] SHEN, H. S., XIANG, Y., and FAN, Y. A novel technique for nonlinear dynamic instability analysis of FG-GRC laminated plates. Thin-Walled Structures, 139, 389–397(2019)
[21] WANG, X., XUE, C. X., and LI, H. T. Nonlinear primary resonance analysis for a coupled thermopiezoelectric-mechanical model of piezoelectric rectangular thin plates. Applied Mathematics and Mechanics (English Edition), 40(8), 1155–1168(2019) https://doi.org/10.1007/s10483-019-2510-6
[22] ZHENG, Y., HUANG, B., YI, L., MA, T., XIE, L., and WANG, J. Nonlinear thicknessshear vibration of an infinite piezoelectric plate with flexoelectricity based on the method of multiple scales. Applied Mathematics and Mechanics (English Edition), 43(5), 653–666(2022) https://doi.org/10.1007/s10483-022-2842-7
[23] ARSHID, E. and KHORSHIDVAND, A. R. Free vibration analysis of saturated porous FG circular plates integrated with piezoelectric actuators via differential quadrature method. Thin-Walled Structures, 125, 220–233(2018)
[24] WANG, K. F., WANG, B. L., XU, M. H., and YU, A. B. Influences of surface and interface energies on the nonlinear vibration of laminated nanoscale plates. Composite Structures, 183, 423–433(2018)
[25] ELMORABIE, K. M., ZAKARIA, K., and SIRWAH, M. A. Nonlinear instability of a thin laminated composite circular plate subjected to a tensile periodic load. Acta Mechanica, 231(12), 5213–5238(2020)
[26] GOLMAKANI, M. E. and MEHRABIAN, M. Nonlinear bending analysis of ring-stiffened circular and annular general angle-ply laminated plates with various boundary conditions. Mechanics Research Communications, 59, 42–50(2014)
[27] MEHRABIAN, M. and GOLMAKANI, M. E. Nonlinear bending analysis of radial-stiffened annular laminated sector plates with dynamic relaxation method. Computers & Mathematics with Applications, 69(10), 1272–1302(2015)
[28] SHOOSHTARI, A. and DALIR, M. A. Nonlinear free vibration analysis of clamped circular fiber metal laminated plates. Scientia Iranica, 22(3), 813–824(2015)
[29] ERSOY, H., MERCAN, K., and CIVALEK, Ö. Frequencies of FGM shells and annular plates by the methods of discrete singular convolution and differential quadrature methods. Composite Structures, 183, 7–20(2018)
[30] MERCAN, K., BALTACIOGLU, A. K., and CIVALEK, Ö. Free vibration of laminated and FGM/CNT composites annular thick plates with shear deformation by discrete singular convolution method. Composite Structures, 186, 139–153(2018)
[31] KARIMI, M. H. and FALLAH, F. Analytical non-linear analysis of functionally graded sandwich solid/annular sector plates. Composite Structures, 275, 114420(2021)
[32] JAVANI, M., KIANI, Y., and ESLAMI, M. R. Geometrically nonlinear free vibration of FGGPLRC circular plate on the nonlinear elastic foundation. Composite Structures, 261, 113515(2021)
[33] HAGHANI, A., MONDALI, M., and FAGHIDIAN, S. A. Linear and nonlinear flexural analysis of higher-order shear deformation laminated plates with circular delamination. Acta Mechanica, 229(4), 1631–1648(2017)
[34] CHAI, Q., WANG, Y., and TENG, M. Nonlinear free vibration of spinning cylindrical shells with arbitrary boundary conditions. Applied Mathematics and Mechanics (English Edition), 43(8), 1203–1218(2022) https://doi.org/10.1007/s10483-022-2892-7
[35] YUAN, T. C., YANG, J., and CHEN, L. Q. Experimental identification of hardening and softening nonlinearity in circular laminated plates. International Journal of Non-Linear Mechanics, 95, 296–306(2017)
[36] CHIANG, D. C. and CHEN, S. S. H. Large amplitude vibration of a circular plate with concentric rigid mass. Journal of Applied Mechanics, 39, 577–583(1972)
[37] RAJU, K. K. Large amplitude vibrations of circular plates carrying a concentrated mass. Journal of Sound and Vibration, 50(2), 305–308(1977)
[38] HUANG, C. L. D. and WALKER, H. S., JR. Non-linear vibration of a hinged circular plate with a concentric rigid mass. Journal of Sound and Vibration, 126(1), 9–17(1988)
[39] HUANG, C. L. D. and HUANG, S. T. Finite element analysis of non-linear vibration of a circular plate with a concentric rigid mass. Journal of Sound and Vibration, 131, 215–227(1989)
[40] LI, S. R., ZHOU, Y. H., and SONG, X. Non-linear vibration and thermal buckling of an orthotropic annular plate with a centric rigid mass. Journal of Sound and Vibration, 251, 141–152(2002)
[41] CHEN, L. Q., LI, X., LU, Z. Q., ZHANG, Y. W., and DING, H. Dynamic effects of weights on vibration reduction by a nonlinear energy sink moving vertically. Journal of Sound and Vibration, 451, 99–119(2019)
[42] LI, X., DING, H., and CHEN, L. Q. Effects of weights on vibration suppression via a nonlinear energy sink under vertical stochastic excitations. Mechanical Systems and Signal Processing, 173, 109073(2022)
[43] WANG, G. X., DING, H., and CHEN, L. Q. Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beams. Journal of Sound and Vibration, 474, 115265(2020)
[44] CHEN, W., HU, Z. Y., DAI, H. L., and WANG, L. Extremely large-amplitude oscillation of soft pipes conveying fluid under gravity. Applied Mathematics and Mechanics (English Edition), 41(9), 1381–1400(2020) https://doi.org/10.1007/s10483-020-2646-6
[45] AMABILI, M. Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass. Journal of Sound and Vibration, 329, 4501–4514(2010)
[46] AMABILI, M. and CARRA, S. Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses. Journal of Sound and Vibration, 331(1), 155– 166(2012)
[47] NAYFEH, A. H. and PAI, P. F. Linear and Nonlinear Structural Mechanics, John Wiley and Sons, New York, 388–392(2004)
[48] INMAN, D. J. and SINGH, R. C. Engineering Vibration, Prentice Hall, Englewood Cliffs, New Jersey, 321–323(1994)